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Melt butter in a large non-reactive saucepan over medium heat.
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Add onions and carrot; sauté until vegetables are softened, about 6 minutes.
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Reduce heat to low. Add flour and cook, stirring, for 4 minutes.
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Gradually add broth, 2/3 cup at a time, cooking and stirring constantly until smooth.
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Stir in stout, thyme, rosemary and bay leaf.
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Bring to a simmer and cook, uncovered, stirring occasionally, until thickened, about
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20 - 25 minutes. Remove from heat.
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Strain mixture through a sieve into a pot.
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Season to taste with salt and pepper.
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Bake the tater tots according to bag instructions.
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Place the tater tots in a baking dish in layers. Add portions of each cheese as you add the tater tot layers.
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Put back in the oven and bake for 5 - 7 minutes until the cheese melts.
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Serve by topping with gravy.
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Enjoy!