Preheat oven to 425°F (220°C). In a large skillet, heat olive oil. Add garlic paste and onion and cook over medium-low heat, stirring occasionally, until onion softens, 15 min. Stir in sugar and salt, then balsamic vinegar. Add beef broth and simmer until liquid evaporates, 5 to 10 min. Stir in thyme leaves, remove from the heat.
Meanwhile, remove the ends of each loaf (discard ends or save for another use), and slice each loaf into 8 pieces. Put a spoonful of the onion mixture onto each slice and sprinkle with cheese.
Bake in the oven on a parchment paper-lined baking sheet until cheese is bubbling and golden, 10 min. Serve immediately.
The topping for this irresistible appetizer can be made the morning of.