Place brisket, carrots, and onions in a large pot.
Add a muslin bag containing parsley, peppercorns, and bay leaf to the pot.
Cover with cold water and bring to a boil before lowering to a gentle simmer.
Simmer uncovered for up to three hours until the meat slides off a fork.
Transfer meat to a container, strain the cooking liquor over it, and let cool.
Discard liquor, cover meat with film, and press under a weighted tray.
Slice the chilled brisket into steak-sized pieces.
Mix all marinade ingredients together in a bowl.
Grill steaks on a hot griddle pan, brushing frequently with marinade.
Whisk together all dressing ingredients.
Blanch broccoli in boiling salted water, then refresh in cold water.
Toss lettuce and broccoli in dressing and serve topped with meat.