Dry brine the rabbit: Pat the rabbit legs dry. Season evenly with the kosher salt and black pepper. Place uncovered on a rack in the refrigerator for 4-8 hours, or plan ahead and brine for 24 hours. This step immensely improves flavor and moisture retention.
Brown the rabbit: Remove the rabbit from the refrigerator 30 minutes before cooking. Preheat oven to 300°F (150°C). Heat a Dutch oven over medium heat. Add the olive oil and butter. Brown the rabbit legs lightly on all sides. You're looking for a light golden color, not a deep brown crust. Transfer the rabbit to a plate.
Build the braise: Reduce the heat slightly. Add the shallots and cook 3-4 minutes until softened. Add the garlic and cook 30 seconds. Add the wine, scraping up any fond, and reduce by about half, 5-7 minutes. Add the chicken stock, thyme, and bay leaf. Return the rabbit legs to the pot — the liquid should come about halfway up the legs.
Braise: Cover and place in the oven. Cook for approximately 1 1/4 to 1 1/2 hours, beginning to check around 75 minutes. The rabbit is done when a knife slides in easily, the meat is tender, and it is just beginning to loosen from the bone. Avoid cooking until it completely falls apart.
Make the pommes purée: While the rabbit braises, cover the potatoes with cold water in a pot, add the salt, and bring to a boil. Simmer until fork-tender, 15-20 minutes. Drain well and return to the pot over low heat for a minute to dry out. Pass through a ricer or food mill back into the pot. Set over low heat and stir in the warm milk, then the cold butter a few cubes at a time, until smooth and glossy. Season with salt and a pinch of white pepper. Keep warm, pressing plastic wrap directly on the surface if holding. (The plastic wrap on the surface of the potatoes keeps a crust from forming.)
Blanch the haricots verts: Bring a pot of well-salted water to a boil. Blanch the haricots verts for 3-4 minutes until crisp-tender and bright green. Shock in ice water, then drain well. Set aside until ready to finish.
Finish the sauce: Transfer the rabbit legs to a warm platter and tent loosely with foil. Discard the thyme stems and bay leaf. Bring the braising liquid to a gentle simmer and reduce until slightly thickened, about 8-10 minutes. Whisk in the Dijon and whole-grain mustard and simmer 1 minute. Whisk in the cream. If too watery, add a little cornstarch slurry. Taste for seasoning, adding salt if needed, and lemon juice or vinegar if the sauce needs brightness. The finished sauce should be silky, mustard-forward, and lightly coating.
Finish the haricots verts: Melt the butter in a skillet over medium heat. Add the shallot and cook 1 minute. Add the blanched haricots verts and toss until warmed through, 2-3 minutes. Season with salt and pepper.
Serve: Place a mound of pommes purée on each plate. Arrange the haricots verts alongside. Lean rabbit leg against the potatoes. Spoon the mustard sauce generously over the rabbit and a little onto the potatoes. Finish with the chopped parsley.