Heat olive oil in a large skillet over medium heat, then add the diced onion and cook until soft, about 4 minutes. Stir in the garlic and cook for 30 seconds until fragrant. Add the black beans, refried beans, chili powder, cumin, and smoked paprika, stirring everything together until warmed through and well combined.
Fold the cooked rice into the bean mixture right in the same skillet. This lets the rice soak up a little of the spiced bean flavor instead of tasting plain inside the burrito. Taste the filling here and adjust the salt and pepper before you move on, since this is your last chance to season it properly.
In a saucepan, melt the butter over medium heat, then whisk in the flour to form a paste-like roux. Cook for about a minute, whisking constantly, until it smells lightly toasted. Slowly pour in the vegetable broth while whisking so no lumps form, then stir in the tomato sauce, sugar, chili powder, and garlic powder.
Bring the sauce to a gentle simmer and let it cook for 8 to 10 minutes, stirring often, until it thickens to a gravy-like consistency. If it gets too thick, splash in a little extra broth. This sauce is what turns an ordinary burrito into a proper smothered burrito, so don't rush this step.
Wrap the tortillas in a damp kitchen towel and microwave for 30 to 45 seconds to make them pliable. Lay each tortilla flat, spoon about ⅔ cup of the bean and rice filling slightly off-center, then fold in the sides and roll tightly, seam-side down.
Preheat your oven to 375°F and lightly grease a 9x13-inch baking dish. Arrange the rolled burritos seam-side down in the dish, nestling them close together so they hold their shape while baking.
Ladle the warm sauce generously over the burritos, making sure every tortilla gets coated. Sprinkle the freshly grated cheese evenly across the top, then bake uncovered for 20 to 25 minutes, until the cheese is melted and bubbling at the edges.
Let the baking dish sit for about 5 minutes once it comes out of the oven. This short rest lets the sauce settle and the cheese firm up slightly, so the burritos hold together better when you serve them instead of falling apart on the plate.