Trim the roots and dark green tops from leeks, leaving 5 to 6 inches of white and light green. Cut leeks in half lengthwise and rinse well. Place leeks in a single layer in a 9x13-inch baking dish, nesting them together. Pour in the chicken broth. Add the garlic and thyme into the broth between the leeks.
Bake in a preheated 425-degree oven for 40 minutes. After about 30 minutes, sprinkle the chicken with 1/4 tsp. each salt and pepper.
Heat oil in a large skillet over medium heat, add the chicken and cook until golden brown, about 5 minutes per side. Remove from the heat.
Remove the leeks from the oven after 40 minutes and nestle the chicken into them. Add cream and the remaining 1/4 tsp. salt to the hot skillet (do NOT turn burner back on). Stir any browned bits and let the cream heat from the warmth of the pan. Pour cream mixture over the leeks and chicken.
Return the baking dish to the oven and bake until chicken is tender, about 20-25 minutes. Serve the chicken with the braised leeks, spooning the sauce over both.