Preheat the oven to 425 degrees . Place the chicken on a foil-lined rimmed baking sheet, rub all over with 2 tablespoons butter and season with salt and pepper. Wrap the garlic cloves in foil and set on the baking sheet. Roast until the chicken juices run clear, about 45 minutes. Set the roasted garlic aside and let the chicken cool for 15 minutes. Discard the skin and bones and finely chop the meat; set aside.
Meanwhile, pierce the sweet potato in several places and place on a paper towel in the microwave oven. Cook on high for 6 minutes. Turn over and cook until tender, 6 to 10 minutes more. Cover with a paper towel and let stand for 5 minutes. Peel the potato and the reserved roasted garlic and transfer to a large bowl; mash and set aside.
In a small saucepan, melt the remaining 5 tablespoons butter over low heat. Add the onion and cook until softened, about 10 minutes. Stir in 1/4 cup flour and cook over low heat, stirring constantly, for 3 minutes. Slowly add the chicken broth, stirring until a paste forms, then cook for 3 minutes, stirring occasionally.
Stir the paste into the reserved mashed potato mixture. Stir in the reserved chicken. Season with salt and pepper. Cover with plastic wrap and refrigerate until firm, about 30 minutes.
Meanwhile, in a small bowl, beat the eggs with 1 tablespoon water. Place the remaining 1/2 cup flour and the bread crumbs on separate plates. Divide the croquette mixture into eight 1/2-inch-thick oval disks. Coat with the flour and shake off any excess. Dip in the egg mixture, letting the excess drip off, then coat with the bread crumbs. Place the breaded croquettes on wax paper.
In a small skillet, heat the oil over high heat until hot but not smoking. Working in 2 batches, fry the croquettes until golden brown, 1 to 1-1/2 minutes per side. Using a slotted spoon, transfer to paper towels to drain. Serve 2 croquettes, with a lemon wedge, per person.