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1
In a large pot, bring the water to a simmer over medium-high heat.
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2
While the water heats, in another large pot, melt the butter over medium heat. When it's melted, add the onions, carrots, celery, mushrooms, bay leaves, thyme, and parsley. Cook until the onions are translucent and the carrots, celery, and mushrooms are soft, about 8 minutes. With a wooden spoon, stir in the cheese bits. Let the cheese and vegetables sit on the bottom of the pot for short periods of time, no longer than 10 seconds; this will allow the vegetables and the cheese to brown the bottom of the pot a little. (You don't want all the vegetables browned, but just the bottom surface needs a little color.) Stir often.
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3
When the vegetables and cheese at the very bottom of the pot show some brown and the cheese is beginning to melt, slowly introduce the simmering water to the pot, stirring in just 1 cup/240 ml to start. Stirring constantly, deglaze the pan's bottom with the hot water to loosen any browned bits. When the pot bottom is clean of any brown, pour in the remainder of the water. Decrease the heat to medium-low and monitor the heat, adjusting the flame so the broth stays at a gentle simmer.
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4
Simmer for 40 to 50 minutes, stirring every 3 to 5 minutes, so the broth doesn't pick up a scorched flavor. Strain the broth into a very large container or another clean pot and allow it to cool. Once it's cool, you can easily skim the top of any fats. Store this in your refrigerator for up to 3 days or in your freezer for up to 3 months.
