In a large, heavy pot or skillet, warm 1/4 cup olive oil over medium-low heat. Add the onions and 1 sprig thyme and cook, stirring every 10 minutes or so (if they start sticking to the pan, add a splash of water), for about 1 1/2 hours. Season to taste with salt and pepper.
Meanwhile, preheat the oven to 350 degrees . Arrange the baguette slices on a baking sheet and brush on both sides with the remaining 2 tablespoons olive oil. Season with salt and pepper. Bake until crisp, 10 to 15 minutes.
Place about 1 tablespoon cheese on each toast; bake until the cheese is melted, about 5 minutes.
Remove the leaves from the remaining thyme sprig and chop. Top each toast with the caramelized onions and chopped thyme.
