There is a suitably ghoulish story behind the restaurant name. In 1626 a society beauty was murdered here by a spurned lover, her dismembered body left on the cobblestones. Londoners love a gory tale, and this story even made it into Charles Dickens novel Little Dorrit; the yard where the crime was committed was named Bleeding Heart Yard after the murder. Since 1983 The Bleeding Heart has been turning out traditional French food in a corner of this yard. The main dining room is downstairs, but there is also a ground floor room, and the original restaurant has spawned a bistro and a tavern under the same ownership just around the corner in Greville Street.
The dining room had a low ceiling, formal white linen and prints of assorted grapes lining the walls. There was a wooden floor, but no muzak, so the noise levels are reasonable. The pictures of grapes are presumably intended to draw attention to the wine list, which was substantial. This weighed in at over thirty pages and ranged far and wide, with even a wine from China listed. The cheapest wine I could spot was £18.75, and the list included choices such as Gimblett Pinot Noir 2208 at £36.50 for a wine that costs £11 in the shops, Trimbach Riesling Reserve 2009 at £47 for a wine that retails at £14, and Confuron Coteidot Chambolle-Musigny 2006 at a chunky £119 for a wine that will set you back £29 in the shops. At the prestige end of the list, Clos St Hune 2002 was a hefty £340 for a wine that retails at £121, and Mommessin 1995 Clos de Tart at an ambitious £675 for a wine that will set you back £218 on the high street.
I opted for the lunch menu at £25, but the a la carte menu had starters at £5.95 to £12.95, main courses at £13.45 to £28.95, with vegetables £3.50 extra. Bread was made from scratch in the kitchen and comprised rolls of white, brown and olive bread. I commend them for making their own bread but the rolls were merely pleasant (14/20). My fish soup was a little thin in terms of taste, but this was bolstered by the traditional parmesan and rouille, and was decent enough (12/20). Red gurnard was accurately cooked and served on a bed of tagliatelle, with strips of bell peppers and tomatoes. The pasta had good texture though the peppers were cooked so long that they were quite soggy (13/20 overall).
My dessert was less successful. Roulade of iced chocolate and hazelnut was served with summer berries, a fruit coulis and a few hazelnuts. Unfortunately, though the fruit was fine the roulade itself was served at such a low temperature that the centre was rock hard, even after several minutes, and the roll around it was also hard and difficult to cut through. This was not something that should have left the kitchen (7/20). Service was polite and the bill came to £32 for the meal and a bottle of water. Other than the dessert, this was competent cooking, but not sufficiently good for me to want to rush back.
@RichW1988 Objectively best is that of Hakkasan/Yauathca. A. Wong for innovative/modern, and Royal China & Princess Garden (good value).