Melt 1 teaspoon butter in a large saucepan over medium heat; add sweet onion and cook until softened, about 4 minutes. Stir garlic into the onion; cook and stir together until garlic is fragrant, about 1 minute.
Pour light beer into the saucepan, bring to a simmer, and cook until onion is completely tender, about 10 minutes.
Stir chicken broth into the beer mixture; bring to a boil. Stream milk into the saucepan while stirring to form the base of the soup.
Heat 1/4 cup butter in a microwave-safe bowl in microwave oven until melted and hot, 30 to 60 seconds. Stir flour into the melted butter until the mixture binds into a roux; pour into the soup. Cook and stir soup until it begins to simmer and thicken, 7 to 10 minutes.
Remove soup from heat. Melt Gouda cheese and Cheddar cheese into the soup a handful at a time, stirring each addition into the liquid until smooth; season with black pepper. Ladle soup into bowls; garnish with bacon and chives to serve.
