Preheat the oven to 425 degrees F. Spread the Tater Tots on a baking sheet and bake, tossing halfway through, until crisp, about 20 minutes. Sprinkle with 1 cup cheese and continue baking until melted, about 5 minutes.
Meanwhile, heat the butter in a large pot or Dutch oven over medium-high heat. Add the onion, season with 1/4 teaspoon salt and a few grinds of pepper and cook, stirring occasionally, until softened, about 5 minutes. Add the garlic and cook until just softened, about 1 minute. Add the potatoes, chicken broth, 3/4 cup water and 1/2 teaspoon salt and bring to a boil. Reduce the heat to a simmer, cover and cook until the potatoes are tender, 10 to 12 minutes.
Meanwhile, cook the bacon in a small skillet over medium-high heat until crisp, 5 minutes. Drain on paper towels.
Puree the soup with an immersion blender, thinning with up to 1/4 cup water. (Alternatively, puree the soup in batches in a blender, then return to the pot.) Stir in 1/4 cup sour cream and the remaining 2 cups cheese until melted. Ladle the soup into bowls. Top with the Tater Tots, bacon, chives and remaining 1/4 cup sour cream. Season with pepper.