Melt enough butter to make 4 tablespoons of fat. Sprinkle in 1/4 cup flour and whisk to combine. Cook for 2-3 minutes, stirring constantly. Whisk in the chicken broth and milk in several intervals alternating between the two.
Add the potatoes, onions, and garlic back to the pot and simmer until the potatoes are tender.
Whisk in both shredded cheeses until melted. Remove from the heat and stir in the bacon, sour cream, chives, and salt and pepper to taste.
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