Combine salt, sugar and 1½ litres water in a large non-reactive container. Submerge pork with weights in brine, cover and refrigerate for 24 hours.
Preheat oven to 175C. Remove pork from brine, rinse and pat dry with kitchen towel, place in a casserole with wine, onion, carrot, leek, garlic, thyme and 750ml water. Cover and cook in oven until tender (1½-2 hours). Drain pork (discard liquid), press between two trays covered with baking paper, weight with food cans and refrigerate until completely cold.
For tomato soup, heat olive oil in a small saucepan over medium heat, add carrot, onion, garlic and shallot, stirring occasionally until soft (7-10 minutes). Add chilli flakes, tomato and bouquet garni. Simmer, stirring occasionally until infused and warmed through (10-15 minutes). Remove from heat, discard bouquet garni, process in a blender until smooth. Stir through crème fraîche, add water to adjust to soup consistency, if necessary. Pass through a fine sieve, season to taste and keep warm.
Meanwhile, for rocket purée, blanch rocket (5-10 seconds), then refresh. Drain well, squeeze out excess moisture, and process in a small food processor until finely chopped (1 minute). With motor running, add oil and process until a fine paste forms, season to taste, refrigerate until required.
Cut pork into 5mm slices. Spread half the brioche slices with rocket purée, layer with pork and truffle, season to taste, top with remaining bread slices. Brush outsides with clarified butter and toast, one at a time, in a large frying pan over medium heat, turning once, until golden (1-2 minutes, each side). Cut into fingers, serve warm with tomato soup.
Note Use female pork belly where available.San Marzano tomatoes are a type of plum tomato. Substitute a canned Roma-style tomato. For the bouquet garni, Thomas Keller uses 1 bay leaf, 1 thyme sprig, 1 tarragon sprig and 2 black peppercorns.