In a large saucepan, melt butter over medium heat. Add garlic, and cook until soft, about 1 minute. Reduce heat to medium-low, stir in flour, and cook, stirring, until it bubbles. Gradually add 1/2 cup stock, about 1/4 cup at a time, stirring constantly to incorporate each addition completely. Slowly add remaining 1 1/2 cups stock and the water, still stirring. Add mushrooms and 1/4 teaspoon salt, and simmer until mushrooms are soft, about 20 minutes.
Add carrots, return soup to a simmer, and then remove from heat. Stir in half the cheese until it melts into the liquid, and then stir in remaining cheese. Serve at once.
When Sarah Carey made this recipe, she replaced the carrots with 2 cups broccoli florets.