Following a soft-launch period, the 40-cover eatery was officially opened this week above champagne bar Amuse Bouche on Parsons Green Lane in the popular area of southwest London.
Dubbed by the team behind the venture as a 'quintessential neighbourhood restaurant', Claude's Kitchen features a weekly-changing menu focused on fresh and seasonal produce which is not readily available in mainstream restaurants.
"The menu encourages diners to try under-used cuts of meat, choose sustainable fish (sent up from Cornish fishing boats the previous day) and learn about lesser-loved British ingredients that work in delicious harmony on the plate," said a statement released by the restaurant.
This week's menu includes Cornish ling with fennel, baby plum tomatoes, new potatoes and Fino sauce as well as Barbary duck leg with red cabbage, currants, lentils and redcurrant gel. The drinks menu features craft beers and a range of cocktails.
Compton has most recently worked at Petersham Nurseries alongside both Skye Gyngell & Greg Malouf. He has also worked at Club Gascon and founded his street food firm Green Goat Food with Jan Smith last summer.
Green Goat Food recently featured at The Restaurant Show where street food was very much in the spotlight - the trend was discussed at the Business Boot Camp sponsored by BigHospitality and street food vendors also catered for visitors to the show.
Also in the kitchen team is sous chef Pete Hartley-Booth who met Compton while the pair were working at Petersham Nurseries.
"All are extremely talented chefs who have worked in a range of Michelin-starred kitchens under the tutorage of some of the best names in the business," said a statement from the restaurant this week.
"This has created a fun and dynamic working environment and an energy which is shown in the cuisine, hospitality and atmosphere of the restaurant."
The restaurant team have also designed a bar menu for the downstairs venue. The restaurant is open from Tuesday to Saturday.