Preheat the oven to 375 degrees . Bring a large pot of water to a boil over high. Salt the water, stir in half the chard; cook until bright green and wilted, 2 to 3 minutes. Using tongs, transfer to a colander. Repeat with the remaining chard. When cool enough to handle, squeeze out excess water.
In a large saucepan, melt the butter over medium. Add the garlic; stir until fragrant, about 30 seconds. Add the flour; whisk 1 minute. Whisking constantly, gradually add the milk. Reduce the heat to medium-low and simmer, stirring often, until the sauce thickens slightly, about 5 minutes.
Reduce the heat to low and gradually whisk in 1 cup Parmesan and 1/2 cup Swiss cheese. Stir in the chard; season with salt and pepper. Transfer to a 3-quart baking dish; sprinkle with the remaining cheese. Bake until bubbling and cheese melts, 15 to 20 minutes.