Place chicken legs, onion, garlic, chicken parts and water in a saucepan. Add 2 sprigs parsley, 1/4 tsp thyme, 1/2 tsp kosher salt, and 1/4 tsp pepper. Bring to a boil, then reduce to a bare simmer; there should be tiny bubbles streaming to the surface, but the surface should barely move (about 170-180°F). Keep at this temperature for 1 hour.
Remove the leg quarters to a plate. Raise the heat and boil the stock vigorously for 20-30 minutes, until it tastes like delicious chicken soup. Drain the broth and discard the solids.
Combine the rice with 1 cup of stock. Bring to a boil, covered, then cook 15-20 minutes, until almost done. Turn off heat and allow the rice to rest.
Meanwhile, in another saucepan, combine peas with 1 lettuce leaf, 1/8 tsp dried thyme, 1/4 tsp sugar and 1 sprig parsley. Place over low heat and cook, stirring occasionally, about 20 minutes. Taste and add salt and pepper if needed; discard the lettuce leaf.
While the rice and peas cook, in the top of a double boiler, combine the egg yolk and vinegar. Whisk constantly while slowly adding 2/3 c of chicken stock. Continue whisking over heat until slightly thickened. Season to taste with salt and pepper. Coarsely chop the remaining parsley and set aside.
Remove the skin and bones from the chicken legs. Shred the meat into bite-sized pieces. Serve the chicken over the rice, sprinkle with parsley to garnish, and pass the sauce on the side. To be very French, serve the peas as a separate course after the chicken, rather than on the same plate.