Set your sous vide machine to 140°F and preheat the water bath.
Season the chicken breasts with salt and pepper, drizzle with olive oil, and vacuum seal in a bag.
Submerge the chicken in the 140°F water bath and cook for 1½ to 2 hours.
Set up the Big Green Egg for direct grilling and preheat to 350°F.
Trim and halve the Brussels sprouts, blanch briefly in boiling water, then toss with olive oil and minced garlic.
Preheat a cast iron pan on the Big Green Egg, add the Brussels sprouts, and cook for 10 minutes, stirring occasionally.
Add the chicken stock to the Brussels sprouts halfway through cooking and continue roasting until tender and lightly caramelized.
Remove the chicken from the sous vide bag and pat completely dry with paper towels.
Place the chicken skin side down over direct heat and grill for about 2 minutes until the skin is crisp and golden brown.
Flip the chicken briefly for 30 to 60 seconds, then remove from the grill and let rest for 5 minutes.
For the gravy, sauté the shallots in butter until softened, stir in the flour, then slowly whisk in the chicken stock.
Simmer the gravy until slightly thickened and season to taste with salt and pepper.
Slice the chicken and serve with the Brussels sprouts and pan gravy.