Preheat oven to 375 degrees. Make the bechamel: Melt butter in a saucepan over medium heat. Add flour and whisk until mixture bubbles slightly but has not started to brown, about 2 minutes. Gradually whisk in milk. Raise heat to medium-high and bring to a boil, whisking often. Reduce heat to low and cook, stirring occasionally with a wooden spoon, until thickened, 12 to 15 minutes. Season with coarse salt and pepper. Remove from heat. Add smoked Gouda and stir until melted.
Meanwhile, blanch the brussels sprouts: Bring a medium pot of salted water to a boil. Cook brussels sprouts until just tender when pierced with the tip of a paring knife, 3 to 4 minutes. Drain and transfer to an 8-cup (8-by-12-inch) baking dish.
Assemble the gratin: Pour bechamel over brussels sprouts and sprinkle with aged Gouda and a pinch of smoked sea salt. Bake, uncovered, until bubbling and golden, about 25 minutes.
Bechamel can be refrigerated for up to 4 days. Cooled blanched brussels sprouts can be refrigerated for up to 1 day. Assembled gratin can be refrigerated overnight. Bring to room temperature before baking.