Preheat oven to 425 degrees. Line a baking sheet with foil or a baking mat, set aside.
Place scrubbed, dried potatoes on the baking sheet. Rub each with oil, sprinkle with sea salt, and prick with a fork three times on each top.
Bake for 1 hour or until fork tender (this could take up to 1 hour 15 minutes). Remove from the oven and allow to cool.
Meanwhile, in a small saucepan set over medium heat, add the butter. When the butter has melted, whisk in the flour, cook for 1 minute.
Slowly add milk, whisking to create a smooth sauce. Bring mixture to a low simmer.
Add salt, pepper, garlic powder, onion powder, and paprika. Simmer until thickened, about 5 minutes.
Remove from heat and stir in cheese, then broccoli.
Remove top of potatoes, then score the inside and remove about 2 tablespoons of potatoes (you can spoon it onto the tops and serve it with it), scoop a generous amount of the sauce inside of each potato. Return the reserved potato and top back. Serve immediately.