Heat the broth in a medium saucepan until simmering.
Melt the butter in an oven-safe braiser (or large cast-iron skillet with a lid or 4-quart saute pan with a lid) over medium-high heat. Add the shallots and mushrooms, tossing to coat in the butter, and cook, stirring once or twice, until golden, about 5 minutes. Stir in the garlic and cook until fragrant, 1 to 2 minutes. Add the sherry, allowing the pan to deglaze. Add the rice and allow it to toast and soak in the flavors of the pan. Add about 4 cups of the hot broth to the pan. Add the thyme and bay leaf. Cover the braiser and place it in the oven to bake until the rice is almost cooked, 1 hour to 1 hour and 15 minutes.
Meanwhile, in a small bowl, mix together the coriander, salt and pepper. Brush the chicken with the olive oil, then dust all over with the seasoning. Set aside.
Fluff the rice with a fork and check on doneness. It should still have some chew with some grains just starting to pop open. Stir in the cranberries. Add the remaining hot broth to keep the dish brothy. Add the chicken breasts to the braiser, laying them at an angle so that most of the chicken is submerged in the rice. Shingle the lemon slices in between the chicken breasts. Cover and bake for 20 to 25 minutes. Uncover and continue to cook until the chicken registers an internal temperature of 165 degrees F, about another 10 minutes. Top with the parsley or thyme before serving.