Melt the butter in a 4-quart soup pot. Add the carrots, onions, celery, and mushrooms and cook, covered, over low heat, stirring occasionally, until the vegetables are tender and lightly colored, 25 to 30 minutes.
Add the remaining ingredients, using enough water to cover the solids, and bring slowly to a boil. Reduce the heat and simmer, partially covered, for 30 minutes, no longer.
Remove the stock from the heat and cool. Pour through a cheesecloth-lined strainer set over a large bowl, and discard the solids.
Taste the stock. If it seems to lack intensity, return it to the pot and boil for another 15 to 20 minutes.
Store the stock, covered, in the refrigerator. Remove any solid fat that rises to the surface before serving or freezing.