Heat the butter in a small cast-iron skillet over medium-high heat. Add the onion and cook, stirring occasionally, until softened and starting to brown, about 8 minutes. Add the garlic and cook, stirring, for 30 seconds. Top the onion mixture with the cheeses. Add the milk and stir everything to combine.
Place the skillet inside a roasting pan or baking dish and fill the pan with enough hot water to come halfway up the sides of the skillet. Transfer to the oven and bake until the cheese is melted and gooey, 20 to 25 minutes. Sprinkle with chives and serve immediately with crackers.