Poach brisket: Trim excess fat from outside of brisket. Put meat in a large pot and add enough water to cover by 1 in. Bring to a simmer over medium heat, skimming any foam from surface. Add onions, celery, carrots, parsley, bay leaves, and tomatoes, crushing tomatoes between your fingers as you add them. Add salt. Cover partially and simmer gently until brisket feels tender when probed with a fork, 3 1/2 to 4 hours. Cool in broth, then chill overnight.
Skim off any congealed fat. Reheat brisket in the broth over medium-low heat.
Make salsa verde: In a bowl, combine oil, parsley, anchovies, shallots, capers, and lemon zest and juice. Stir in bread crumbs and egg and season with salt.
Transfer brisket pieces to a cutting board. Using a very sharp knife, cut meat against the grain into 1/2-in.-thick slices, then into 2-in. pieces. Arrange pieces in a single layer on a platter and drizzle with a little of the hot broth to moisten them. Dollop salsa verde onto meat. Serve immediately, with extra salsa on the side.
Make ahead: Brisket, up to 2 days through step Salsa, up to 4 hours, covered and chilled.
Note: Nutritional analysis is per serving.