Heat the oven to 450° F and position a rack in the middle of the oven.
Scrub the potatoes well, then rinse and dry them. Rub the skins with a little olive oil and sprinkle all over with salt. Poke several holes in the potatoes, using a fork with sharp tines. Put the potatoes directly on the oven rack and bake until completely tender, 45 to 60 minutes.
Remove the potatoes from the oven, and being careful not to burn your fingers, split them down the middle with a sharp knife. Scoop the insides of the potato into a bowl, leaving the skins intact. Mash in the broccoli, creme fraiche, 1 tablespoon of the butter, 1/2 cup of the cheddar, and salt and pepper to taste.
Pile this mixture into the skins and transfer the potato halves to a baking sheet. Dot the tops of the potatoes with the remaining tablespoon of butter and sprinkle evenly with the rest of the cheese. Bake for a second time until golden, 10 to 15 minutes. (If you want to turn on the broiler for the last few minutes to get the tops nice and brown, I say go for it.)