Melt 1 tablespoon butter in a skillet over medium heat. Add broccoli and onion and cook until softened, about 5 minutes. Remove from heat and let cool.

Distribute sauteed vegetables evenly on 2 slices of bread. Place 2 slices Cheddar cheese over the vegetables, season with garlic powder, and top with remaining 2 slices of bread.

Melt 1 tablespoon butter in a nonstick skillet over medium-high heat, tilting the skillet to distribute the butter evenly as it melts. Place the sandwiches into the skillet and cook until the bottom piece of bread is golden brown, about 5 minutes.

Melt remaining butter in the skillet. Carefully flip each sandwich, evenly coating the bread with melted butter. Cook until each sandwich is golden brown, 2 to 5 minutes more.