Preheat oven to 450°. Season chicken with salt. Heat a large, heavy ovenproof skillet (preferably cast iron) over medium-high heat. Swirl in oil. Cook chicken, skin side down, until browned, about 5 minutes. Flip chicken, and push to side of skillet. Add 1 cup stock and 1/2 teaspoon salt. Add potatoes to liquid. Bring to a boil. Add garlic, olives, lemon wedges, and thyme to liquid. Return to a boil.
Heat broiler. Preheat a large, heavy ovenproof skillet (preferably cast iron) on top rack for 15 minutes. Meanwhile, toss together tomatoes, beans, capers, garlic, 1 tablespoon oil, and 1/2 teaspoon salt in a bowl. Toss shrimp with remaining tablespoon oil and 1/4 teaspoon salt in another bowl.
Remove skillet from oven, and add tomato mixture. Broil until tomatoes are charred and beginning to collapse, about 5 minutes. Remove from oven, and stir in stock, then shrimp.
Broil, stirring halfway through, until shrimp are opaque, about 3 minutes more. Drizzle with oil.
Iron is one of the more reactive metals, which is why traditional wisdom is not to cook tomatoes and other acidic foods in cast iron, since they can pick up a metallic flavor. In our experience, this isn't an issue if the pan is well seasoned.