Make caramelized onions: In a large Dutch oven (7 quarts/6.7 liters), heat butter over medium to medium high heat.
Add onions and shallots in 3 batches. First batch: toss ⅓ onions in butter. Cover and cook for 3 minutes, stirring occasionally.
Second batch: Toss another ⅓ of the onions and garlic. Add ½ cup white wine. Cover and cook for 6 minutes, stirring occasionally.
Third batch: toss final ⅓ of the onions and add salt and pepper. Add ½ cup wine. Cover and cook for 6 minutes. *If needed, adjust temperature back to medium heat for this 3rd step as onions become more dense. You want to ensure a steady course of heat to wilt the onions.
Remove lid after 6 minute mark (3rd batch). Continue to cook with lid off, stirring occasionally, about every 3-4 minutes, until onions begin to stick to bottom of pot, and begin lightly browning, for another 40-45 minutes. The total caramelization process from start to finish should take about 1 hour to 1 hour 10 minutes depending on your heat consistency.
Add remaining ¾ cup wine. stir and continue to cook for another 2 minutes until slightly reduced.
Add stocks. Toss in thyme and stir. Bring to a light boil and then reduce heat.
Cover, simmering over low heat for 35-40 minutes. Stir occasionally.
Serve warm with croutons and additional cheese to finish.