Preheat oven to 425 F. Prick potatoes with fork four or five times each for steam to vent and place directly on middle rack in oven; bake 40 to 60 minutes, until soft when squeezed using an oven mitt. In a medium pot, melt 1 tablespoon butter over medium heat. Add onion and sauté until tender, 5 to 7 minutes. Add carrots and cook another minute or two. Stir in broth, onion powder, garlic powder, pepper, and cayenne. Bring to a simmer, then add broccoli; cover and cook 3 to 5 minutes, until tender.
Meanwhile, in a medium saucepan, melt remaining butter over medium heat. Whisk in flour and cook 1 to 2 minutes, whisking constantly until golden brown with a nutty scent. Whisk in milk; continue whisking until thickened and bubbly, about 5 minutes. Whisk in cheese a handful at a time until fully melted. Remove from heat and stir cheese sauce into pot with broccoli. Cover and set aside. Split potatoes open with knife and fluff insides with fork. Generously top with broccoli cheddar.
Per serving: 491 calories, 21 grams fat (12 grams saturated fat), 63 grams total carbohydrates, 57 milligrams cholesterol, 556 milligrams sodium, 10 grams fiber, 17 grams protein.
