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Step 1
Preheat oven and bring a large pot of salted water to a boil. Preheat oven to 375 degrees F and bring a large stockpot of salted water to a boil.
Step 2
Blanch the Brussels sprouts and potatoes. Halve Brussels Sprouts and slice potatoes into 1/4" rounds. Add the Brussels Sprouts and potatoes to the pot and boil for 3-4 minutes (do not overcook or the potatoes will not maintain their shape and the Brussels sprouts will lose their bright green color). Drain and immediately transfer to a large pot filled with ice water to stop cooking and set the color of the vegetables. When the sprouts are cooled, drain. Set aside.
Step 3
Make roux. Meanwhile, melt butter in a large skillet over medium-high heat. Add the minced shallots and sauté for 3-4 minutes. Add the fresh thyme, rosemary, and garlic and sauté for 1 minute more. Reduce heat to medium and sprinkle the flour on top of the vegetables. Stir to combine, taking care not to burn the flour. Add the marsala wine and whisk well to combine. Cook for one more minute.
Step 4
Finish the sauce. Add the milk and half-and-half to the skillet and whisk well until the sauce is smooth and thickened. Remove from heat and mix in half of the shredded cheese until completely smooth and melted. Season with salt and pepper. Add the Brussels Sprouts and potatoes to the cheese sauce and mix well to combine.
Step 5
Transfer to a gratin dish and bake. Transfer to a greased 8x10 baking dish and cover loosely with foil. Bake for 30 minutes. Sprinkle with fried onions and the remaining shredded cheese and put under the broiler until the cheese is brown and melted.