Heat oil in a large saucepan over medium heat until shimmering. Add the onion and cook until softened and lightly browned, 5 to 7 minutes. Stir in the farro and cook until the grains look lightly toasted and smell nutty, about 2 minutes. Pour in the broth and water and bring to a simmer. Reduce the heat to low and continue to simmer, stirring often, until the farro is tender and the remaining liquid has thickened into a creamy sauce, 20 to 25 minutes. Remove from the heat, stir in the parmesan, and season with salt and pepper to taste.

Bring 4 quarts of water to a boil in a Dutch oven while the farro simmers. Add the broccoli rabe and salt and cook until the greens are just tender and bright, about 2 minutes. Drain the broccoli rabe thoroughly and transfer to a bowl.

Wipe the now-empty Dutch oven dry, add the remaining oil, garlic, and pepper flakes, and cook over medium heat until the garlic is fragrant and sizzling, 1 to 2 minutes. Stir in the drained broccoli rabe and toss until hot and well coated. Remove from the heat and fold in the beans, farro mixture, and sun-dried tomatoes. Season with salt and pepper to taste.

Position an oven rack 6 inches from the broiler element and heat the broiler. Transfer the mixture into a 3-quart broiler-safe casserole dish and sprinkle with the remaining parmesan. Broil until the cheese is lightly browned and the edges are bubbling, 3 to 5 minutes. Let the gratin cool for 5 minutes before serving to allow the sauce to set.