Stuff and cook the snapper: Preheat the oven to 325 degrees F. If the fins are still on the fish, use kitchen scissors to cut them off (they just fall apart when cooked). Sprinkle both the inside and the outside of the fish liberally with salt and pepper. Stuff the cavity with 1/4 teaspoon crushed red pepper, whole basil leaves, oregano sprigs, garlic and lemon slices.
Using a very large (14-inch) saute pan or a small roasting pan (oven safe) heat about 2 tablespoons of the olive oil over medium-high heat until smoking. Add the fish and cook until well browned on one side, 3 to 4 minutes. Flip the fish over-use a large spatula and grab the tail if it's not too hot-and cook on the other side until well browned, about 2 minutes. Take the fish out of the pan and use wadded paper towels to wipe the pan clean of the oil.
Cook the vegetables: Add another tablespoon of olive oil to the pan, a pinch of red pepper flakes, the tomatoes, potatoes, chopped oregano and a pinch of salt and cook until the tomatoes begin to lose some of their liquid, about 3 minutes. Return the fish to the pan, laying it on top of the vegetables and drizzle about a tablespoon of olive oil over it. Add the chicken broth to the pan and squeeze the remaining lemon half into the sauce in the pan. Cook on the stove another 3 to 4 minutes, basting the fish with the sauce and moving some of the tomatoes and potatoes in the pan on top of the fish.
Finish cooking the fish in the oven safe pan in the oven 18 to 19 minutes. (To check for doneness, push the head down gently in the area between the nose and the top of the head; when the head easily moves away from the back of the neck, it is ready.)
To serve: Remove the fish from the oven and carefully transfer it to a sided platter, pouring any liquid in its cavity back into the pan, and keep the fish warm. Return the pan to the stove over medium heat and cook the sauce and vegetables for a minute or two to concentrate the flavors. Add another tablespoon of olive oil to the pan, the chopped basil and season with additional salt and pepper if necessary. Pour the sauce and vegetables over the fish and serve, using a large spoon to remove servings off the fish. (Don't forget to turn it over to eat off the other side!)