Over a medium flame, melt the butter in a saucepan. Sprinkle the flour in - gradually. Stir constantly with a whisk, until the roux turns golden-brown in color and emits a nutty aroma. Lower the flame and introduce carrot, onion, garlic, and peppercorns into the pan. Saute the vegetables until soft, about 8 minutes. Pour in the stock and wine. Whisk until the roux is perfectly incorporated into the liquids. Stir in the tomato paste. Throw in the thyme. Lower the flame to a slow simmer and cook until the sauce is reduced to 1 cup - approximately 45 minutes.
Skim whatever scum and detritus rise to the surface - periodically. Strain the sauce through a China Cap or a fine mesh collander into a bowl. (Do not press hard on the residue in the collander while straining the sauce. A very light pressing is what is needed.)
Clean and slice the mushrooms. In a 10 inch skillet over a medium flame, saute the mushrooms in 1 Tbl of butter until golden brown.
Place one cup of the Sauce Espagnole into a sauteuse or other sauce pan. Pour in the wine. Place the sauteed mushrooms into the sauteuse. Reduce the sauce until it coats the back of a spoon. Check for seasoning, adjusting as necessary. Cut off heat. Whisk 2 Tbl of cold butter into the sauce.