Preheat oven to 350 degrees F.
Place baguette slices on a baking sheet and brush with about 2 tablespoons olive oil. Bake until lightly toasted, about 5 minutes. Set aside.
For the onions, heat 1 tablespoon olive oil in a large skillet over medium heat.
Add the onions and cook until softened, about 15 minutes.
Stir in brown sugar. Cover and cook until the onions are golden brown, about 20 minutes, stirring occasionally.
Add ¼ cup water, cover the pan again and cook until the onions are deep golden brown, about 15 minutes.
Stir in balsamic vinegar, salt, sage and an additional ¼ cup water. Cover the pan and cook until the liquid is absorbed. Remove from heat.
Using a vegetable peeler, shave off Pecorino Romano slices, one for each baguette slice.
Divide the caramelized onions between the baguette slices and top each one with a slice of cheese.
Place in oven and bake until cheese is beginning to melt, 3 to 4 minutes. Serve.