Start by caramelizing the onion: Heat 1 tablespoon of olive oil in a skillet over medium heat. Add the onions, a pinch of salt, and cook slowly, stirring often, until golden and soft, about 20–25 minutes.
Add brown sugar and balsamic vinegar to the onions. Stir and cook for another 2–3 minutes. Remove from heat and set aside.
Dip each goat cheese round in beaten egg, then coat with breadcrumbs. Chill for 10 minutes.
Heat remaining olive oil in a pan over medium heat. Fry the goat cheese rounds until golden brown on both sides. Remove and place on paper towels.
Toast baguette slices until golden.
To assemble, top each toast with a spoonful of caramelized onion and a piece of fried goat cheese. Garnish with fresh thyme if desired. Serve warm.