Heat oven to 400°F. Butter a 9×13-inch baking dish.
Hasselback the potatoes: set two chopsticks alongside each potato and slice crosswise into thin cuts without cutting through.
Fan the slices gently. Stand the potatoes upright in tight rows in the dish, tucking end pieces into gaps.
Warm cream, milk, melted butter, garlic, thyme, mustard, salt, pepper, and paprika in a saucepan until steaming. Do not boil.
Pour about half the hot cream mixture over the potatoes. Wiggle the rows so it seeps between slices.
Cover the dish tightly with foil and bake 30–35 minutes, until the potatoes start to soften.
Uncover. Pour the remaining cream mixture over the potatoes. Sprinkle all the Gruyère and Parmesan evenly, nudging some between slices.
Bake uncovered 25–30 minutes, until tender, bubbling, and golden on top.
Broil 1–2 minutes for extra color if desired, watching closely.
Rest 10 minutes to set. Shower with chives and serve warm.