1In a large, deep skillet over medium-high heat, melt 3 tablespoons of the butter. Add the onions and stir until they are fully coated. Cook for 10 minutes, stirring frequently, until the onions are softened. Reduce the heat to medium-low and continue to cook until the onions begin to brown. Reduce the heat to the lowest setting and continue cooking the onions until they are very brown. The onions are done when they are uniformly brown and very soft and the volume is about one-fifth of the original amount. This will take approximately 1 hour total, but it’s worth it.
2Increase the heat to medium-high and add the remaining tablespoon butter to the onions. Once the butter has melted, stir in the flour and then cook for 2 minutes. Slowly add 2 cups of the stock and simmer until thickened, approximately 5 minutes (see Note). Add the remaining 4 cups stock and the wine, vinegar, salt, pepper, and thyme. Bring to a simmer, lower the heat, and cook for 5 minutes. Taste for seasoning and add additional salt and pepper as needed.