Pat rabbit dry with a paper towel. Sprinkle salt, garlic powder, paprika, and pepper all over rabbit.
In a 12-inch cast-iron skillet, heat oil over medium heat. Add rabbit, and cook until browned, 4 to 5 minutes per side. Remove from skillet. Wipe skillet clean.
In same skillet, melt 2 tablespoons butter over medium heat. Add onion and mushrooms; cook, stirring frequently, until tender and golden brown, 8 to 10 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute. Remove onion mixture from skillet.
In same skillet, melt remaining ⅓ cup butter over medium heat. Add flour; cook, stirring constantly, until mixture is the color of dark peanut butter. Add broth and thyme, stirring until combined. Add onion mixture and rabbit, stirring until combined. Cover skillet with lid or foil.
Bake until rabbit is tender, about 1 hour. Remove rabbit from skillet, and place on desired serving platter.
Season onion and mushroom gravy with salt and pepper to taste. Pour over rabbit. Serve with rice.
Recipe by Louisiana Cookin' at https://louisianacookin.com/smothered-rabbit-and-rice/