Heat oven to 425°F. Scrub potatoes, pat dry, and prick all over with a fork.
Rub potatoes with olive oil and a pinch of salt. Bake on a rack or sheet until very tender, 50–60 minutes.
While potatoes bake, steam broccoli until crisp-tender, 4–5 minutes. Drain well and chop into small pieces.
Cool potatoes 5–10 minutes. Halve lengthwise and scoop flesh into a bowl, leaving sturdy 1/4-inch shells.
Mash potato with butter, sour cream, and warm milk until creamy. Season with garlic, remaining salt, and pepper.
Fold in chopped broccoli, scallions, and 1 1/4 cups cheddar, mixing just to combine.
Spoon filling back into potato shells. Top with remaining cheddar and sprinkle paprika.
Return to oven and bake until hot and cheese melts, 10–12 minutes. Let rest 5 minutes before serving.