ESPAGNOLE: Fold 1 bay leaf , 3 to 4 sprigs fresh thyme , 3 to 4 fresh parsley stems , and 7 to 8 whole black peppercorns in a square of cheesecloth and tie the corners with a piece of kitchen twine. Leave one string long enough so that you can tie it to the handle of your pot to make it easier to retrieve.
In a heavy-duty medium saucepan, melt 1 ounce ghee over medium heat until it becomes frothy.
Add the mirepoix — ½ cup diced onions , ¼ cup diced carrots , and ¼ cup diced celery —and sauté until lightly browned, about 6 minutes. Don't let it burn.
With a wooden spoon, stir 1 ounce all-purpose flour into the mirepoix a little bit at a time until it is fully incorporated and forms a thick paste (this is your roux).
Lower the heat to low and cook the roux until it starts to take on a very light brown color, 4 to 5 minutes. Again, don't let it burn.
Using a wire whisk, slowly add 3 cups beef stock and 2 tablespoons tomato purée , whisking vigorously to make sure it's free of lumps.
Bring to a boil, lower the heat to low, and add the sachet. Simmer until the total volume has reduced by ⅓ (you'll have about 2 cups), stirring frequently to make sure the sauce doesn't scorch at the bottom of the pan, 35 to 45 minutes.
Use a ladle to skim off any impurities that rise to the surface.
DEMI-GLAZE: Reduce the stock and sauce: In a large heavy bottomed pot, combine 2 cups beef stock with the espagnole sauce. Bring to a boil over medium-high heat. Adjust the heat to a simmer, and cook uncovered for 45 to 50 minutes, or until the sauce has been reduced by half. In the end you should have about 2 cups of sauce.
Strain the sauce: Remove the herb sachet. Set a fine mesh sieve lined with cheesecloth over a medium bowl and strain the sauce.
MUSHROOM SAUCE: In a heavy-bottomed saucepan, heat the butter over medium heat until it is frothy. Add the mushrooms, sauté for 3 minutes.
Then add shallots and sauté for 2 minutes until the mushrooms are soft and the shallots are translucent.
Add the demi-glace, bring to a boil, then lower heat to reduce the sauce, cooking at a simmer for about 10 minutes.
Stir in the sherry, season to taste with lemon juice, and serve right away.
