Saute the carrots slowly in 2 ounces of butter until tender. Chop coarsely and reserve in a large mixing bowl.
Over high heat, saute the mushrooms in 1-ounce of butter for 2 minutes. Chop coarsely and add them to the carrots.
Saute spinach in 1-ounce of butter. Chop coarsely and reserve separately. When the spinach is cool, add 1 egg and mix thoroughly.
Beat together the remaining eggs and the cheese. Combine this mixture thoroughly with the carrots and mushrooms. Add salt and pepper. Taste and correct seasonings, if necessary.
Preheat oven to 400 degrees F.
Line the bottom of an 8 1/2 by 4 1/2 by 2 1/2-inch loaf pan with parchment paper. Butter the pan and the parchment with the remaining 1-ounce of butter. Fill the pan with half the carrot mixture, cover with the spinach, and top with the remaining carrot mixture. Top with another piece of buttered parchment paper
Place in a bain-marie, and bake in oven for 1 hour and 15 minutes, or until a knife inserted into the center comes out clean.
Invert onto a warm serving platter and remove the foil. Slice and serve immediately.