In a large Dutch oven, melt 2 tablespoons of butter over medium-low heat. Add the sliced onions and brown sugar. Stir to combine. Cook, stirring occasionally, until the onions are deep golden brown and caramelized, about 45 minutes to 1 hour. If the onions begin to stick, add a splash of beef broth to deglaze the pot.

Add the minced garlic and fresh thyme leaves to the pot. Stir and cook for 1-2 minutes until fragrant.

Sprinkle the flour over the onions and stir to coat evenly. Cook for an additional 2 minutes to eliminate the raw flour taste.

Pour in the Irish stout beer, scraping the bottom of the pot to release any browned bits. Add the beef broth, apple cider vinegar, stone-ground mustard, bay leaves, salt, and pepper. Stir to combine. Bring the mixture to a boil, then reduce the heat and let it simmer uncovered for 30 minutes.

While the soup simmers, preheat your oven's broiler. In a skillet, melt the remaining 1 tablespoon of butter over medium heat. Add the rye bread cubes and thyme leaves, tossing to coat. Toast the bread until golden brown on all sides, about 5-7 minutes.

Ladle the hot soup into oven-safe bowls or crocks. Top each bowl with the toasted rye croutons and evenly sprinkle the grated Irish cheddar cheese over the croutons. Place the bowls on a baking sheet and broil for 3-5 minutes, or until the cheese is melted and bubbly.

Carefully remove the bowls from the oven and let them cool slightly before serving.