Step 1 In a medium, heavy-bottom saucepan, heat the olive oil over medium heat until hot. Add the celery, onion, carrot and garlic, and cook, stirring frequently, until softened and lightly caramelized, about 10 minutes.
Step 2 Stir in the mushrooms and continue to cook, stirring frequently, until the mushrooms have rendered their liquid and are lightly browned, about 8 minutes.
Step 3Stir in the tomato paste and continue to cook, stirring frequently, until the paste darkens slightly and forms a thin film on the bottom of the pan (careful not to burn), about 5 minutes.
Step 4Stir in the mushroom or vegetable broth, scraping the flavoring from the bottom of the pan, then stir in the tarragon and bay leaf. Continue to cook, stirring frequently, until the mixture is reduced to a thick, sauce-like consistency, 3 to 5 minutes. Season with one-half teaspoon salt and one-fourth teaspoon pepper, or to taste.
Step 5Remove from heat and discard the tarragon and bay leaf. This makes about 2 cups of sauce. Reserve the sauce in a warm place, or cover and refrigerate until needed, warming on the stove top and thinning as needed with additional broth before serving.