Preheat the oven to 350°. In a saucepan of boiling salted water, cook the brussels sprouts until they are just tender, about 5 minutes. Drain and pat dry.
Meanwhile, heat the olive oil in a large skillet. Add the onion, season with salt and pepper and cook over moderate heat, stirring occasionally, until just softened, about 5 minutes. Add the paprika, cover and cook over low heat, stirring frequently, until the onion is tender, about 10 minutes; add a little water if the onion dries out.
Add the brussels sprouts to the skillet and cook until tender throughout, about 5 minutes. Season with salt and pepper.
Arrange the toasts on a baking sheet and top with the cheddar. Bake for 2 minutes, until the cheese is melted; mound the brussels sprouts and onions on top and serve. If you like spicy foods, I like to add some crushed red pepper flakes on top.
**Optional: Add 1 cup sliced or canned mushrooms and cook with the onions. Cook down so they become moist. The mushrooms add some additional sweetness to the bitter brussel sprouts.