Preheat your oven to 375°F. Generously butter an 11-inch pie plate or similar baking dish. Place the dish on a baking sheet or drip pan to catch any overflow.
In a large skillet over medium heat, melt the 4 tablespoons of butter. Add the diced onion and cook for about 5 minutes, until softened. Stir in the minced garlic and cook for another minute, until fragrant.
Whisk in the 1/4 cup flour and cook for 1–2 minutes, stirring often, to cook out the raw flour taste.
Slowly pour in the 4 cups of half-and-half, about 1 cup at a time, whisking constantly. Continue cooking and whisking until the sauce thickens and is smooth and creamy.
Stir in the herbs de Provence, dried mustard, black pepper, and 1 teaspoon salt. Reduce the heat to low and add 1 cup cheddar, 1 cup smoked Gouda, and the Boursin, stirring until all the cheese is melted and the sauce is smooth.
Using a mandolin, thinly slice the 3 pounds of Yukon gold potatoes. Arrange about one-third of the slices in an even layer in the buttered dish. Lightly sprinkle with salt, then pour about one-third of the cheese sauce over the top. Repeat with two more layers of potatoes, salt, and sauce, ending with sauce.
Cover the dish tightly with foil and bake at 375°F for about 1 hour, or until the potatoes are mostly tender when pierced with a knife.
Remove the foil and sprinkle the top with the remaining 1 cup cheddar and 1 cup smoked Gouda. Return to the oven and bake uncovered for 20–30 minutes, or until the top is golden brown, bubbly, and slightly crisp around the edges.
Remove from the oven and let it rest for 10–15 minutes before serving, which allows the sauce to thicken and set.