Heat a large skillet with sides over medium-high heat. Add 4 tablespoons butter and allow the butter to brown, stirring often about 3-5 minutes. Add the garlic and cook for 15-30 seconds or until fragrant, then slowly pour in the milk, heavy cream, and pumpkin puree. Whisk until smooth. Bring the sauce to a gentle simmer and cook for 5-10 minutes. Stir in the pecorino Romano, 1/2 cup Swiss-style cheese, and nutmeg. Season with salt + pepper.
Boil the fettuccine until al dente. Drain and add directly to the pumpkin Alfredo sauce. Cook until warmed through, about 5 minutes.
Meanwhile, heat a skillet over medium heat and add 2 tablespoons butter. Add the Swiss-style cheese slices and cook 2-3 minutes per side or until a crust forms and the cheese has "fried". Remove the cheese from the skillet and add the sage, fry for 30 seconds. Remove the skillet from the heat.
Divide the fettuccine among plates and top with fried cheese + sage. Drizzle with any browned butter left in the pan. Eat!