French Onion Croque Madame

Name: French Onion Croque Madame
Ingredients: 4 yellow onions, thinly sliced; 2 garlic cloves, unpeeled; 4 sprigs thyme; 3 tablespoons olive oil; 2 tablespoons brandy; 1 tablespoon maple syrup; Kosher salt; Freshly ground black pepper; 1 teaspoon white wine vinegar; 1 1/2 cups milk (375 milliliters); 2 tablespoons butter; 2 tablespoons flour; 1 teaspoon kosher salt; 1/4 teaspoon freshly ground black pepper; 1/2 teaspoon herbes de Provence; Pinch freshly grated nutmeg; 1/4 cup finely grated Parmigiano-Reggiano cheese (25 grams); 1 1/4 cups grated Gruyère or Emmental cheese, divided (150 grams); 8 slices sourdough multigrain rye, toasted, such as Stonemill; 4 teaspoons Dijon mustard; 8 thin slices ham, optional (about 150 grams); 4 eggs; Butter, for frying eggs; Fresh thyme, for garnish