In a 6-quart heavy-bottomed stock-pot, heat the oil over medium-high heat. Add the mushrooms, carrot, onion, and shallot and cook until the vegetables are a deep golden brown.
Add the flour and cook for 5 to 6 minutes, stirring to prevent sticking.
Add the garlic, parsley, rosemary, tarragon, bay leaves, stock or water, red wine, tomato paste, tomato, and salt and pepper. Simmer for 1 1/2 hours, stirring occasionally, until reduced by halt. Adjust the seasoning and strain.
For each cup of strained wine sauce, add 2tablespoons of the barbecue sauce. This sauce may be made several days in advance and kept refrigerated.
Note: This sauce may be made several days in advance and kept refrigerated.