Beche began his motorsport career in karting before transitioning to single-seaters in Asia in 2007, finishing fifth overall in the Asian Formula Renault Challenge with one win. In 2008, he was runner-up to Felix Rosenqvist in Formula Asia 2.0.
Beche switched to sportscar racing in 2009, finishing third in Formula Le Mans. In 2010, he competed in four Le Mans Series rounds – three in the FLM class and one in GT1 in a Ford GT – and also entered two rounds of the FIA GT3 European Championship, also driving a Ford. He also drove in the FIA GT1 World Championship round at Zolder in a Ford.
In 2011, Beche competed in the full Le Mans Series season in an LMP2 Oreca 03 for TDS Racing with Pierre Thiriet and Jody Firth, winning at Spa and Estoril and finishing fourth in the drivers' standings. For 2012, LMP2 became the top class of the renamed European Le Mans Series. Partnering Thiriet at TDS, Beche won the opening round at Paul Ricard and the season finale at Road Atlanta, securing his first major championship victory. At the 24 Hours of Le Mans in 2012, they joined Christophe Tinseau, finishing second in class.
Beche remained with Thiriet by TDS in 2013, taking two victories alongside Thiriet, leading the team to second place in the ELMS standings. He also progressed to the FIA WEC's top class, piloting a Lola B12/60 for Rebellion Racing, scoring two podium finishes and finishing fifth in the championship.
Rebellion’s LMP1 programme became Beche’s permanent home for the next two seasons. He achieved an LMP1-L class win at the 24 Hours of Le Mans in 2014 with Nico Prost and Nick Heidfeld, and the eventual LMP1-L title. The team withdrew from the opening two races of the 2015 season to finalise work on their new Rebellion R-One, but once they entered, Beche and Prost took two privateer class wins to become champions. At the end of the year, Beche was part of a shootout for a reserve seat at Toyota Gazoo Racing, though that seat eventually went to Kamui Kobayashi.
In 2016, Beche returned to the ELMS with Thiriet by TDS, driving an Oreca 05 alongside Pierre Thiriet and Ryō Hirakawa. He took pole for the opening race at Silverstone, though an early crash forced the team out. The trio bounced back by winning at Imola and the Red Bull Ring. Beche then took a convincing pole at Le Castellet, paving the way for the outfit's third successive victory. Another podium finish at Spa was not enough to win the title, as the team suffered electrical troubles in the final race.
Beche returned to Rebellion on a full-time basis in 2017, competing in the LMP2 class of the WEC. His #13 entry was hampered by disqualifications at Le Mans, where a miraculous overall podium (and second place in class) was taken away due to “unnecessary modification of approved bodywork”, and Fuji, which saw a drive time violation. Despite taking three podiums from the final four races, he finished fourth in the teams' standings.
For the 2018–19 WEC “Super Season”, Beche and Rebellion returned to LMP1 with the R13, partnering Thomas Laurent and Gustavo Menezes. The team started off with two third places, including a class podium at Le Mans, before a disqualification for both Toyotas earned Beche and his teammates victory at Silverstone. He moved to the sister car for the round at Sebring and left the team with two races to go, finishing sixth in the championship.
After a scattergun racing programme in 2019 and 2020, where he made his Super GT debut for arto Ping An Team Thailand, Beche returned to the rostrum in 2021, winning the final race of the Le Mans Cup and the LMP2 Pro-Am class at the 8 Hours of Portimão with Realteam Racing.
Beche drove for TDS Racing x Vaillante in the ELMS in 2022, finishing fourth in the Pro-Am standings and fourth at Le Mans. He switched to Nielsen Racing for the 2023 ELMS season, finishing fourth in the Pro-Am classification.
Going into 2024, Beche teamed up with Grégoire Saucy and Rodrigo Sales at Richard Mille by TDS in the ELMS. A strong performance in Le Castellet gave the TDS crew victory. Despite his pace at Le Mans, penalties for the team left him ninth in LMP2. Third in Imola allowed the team to take the championship lead. A mid-race crash at Spa-Francorchamps however ended TDS's race early. Beche and his teammates recovered to win in Mugello, putting TDS just two points behind leaders AF Corse ahead of the Portimão finale. An overtake by Matthieu Vaxivière in the final race dropped TDS to fourth in the championship.
During the 2024-25 winter, Beche partnered Jonas Ried and Alexander Mattschull at Proton Competition in the Asian Le Mans Series. He was initially announced to drive a full season in the 2025 IMSA SportsCar Championship for PR1/Mathiasen Motorsports, but was replaced by Benjamin Pedersen before the fourth race of the season at Mosport.
Beche remained at TDS Racing for the 2025 ELMS season, partnering Sales and Clément Novalak. With a win at the season finale at Portimão, Beche and his teammates finished third in the LMP2 Pro-Am standings. He received the Goodyear Golden Wingfoot Award in his category, recognizing his average stint pace across the season. He also scored pole position, fastest lap and second place at the 24 Hours of Le Mans.
As early as 2016, Beche has been described as an “ace” in the LMP2 class.
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