In a medium saute pan, melt 1/4 cup butter over medium heat. Add sliced mushrooms; cook and stir until soft. Remove mushrooms from pan, and set aside.
Add 1 tablespoon butter to pan, and melt . Add shallots; cook and stir until translucent. Add mushrooms, and cook until soft. Add thyme, bay leaf, and red wine. Reduce.
In a small bowl, dissolve arrowroot in 1/4 cup cold beef broth. Stir remaining broth into sauce, and bring to a boil. Whisk in arrowroot mixture, and stir until thick. Add reserved mushrooms. Season to taste with freshly ground black pepper and salt.