Preheat oven to 450°. Using a small melon-baller (3/4- to 1-in. diameter), scoop out as many potato balls as possible from 2/3 of russet potatoes and 2/3 of Yukon Golds. Drop potato balls into a bowl of water as you shape them, to prevent browning.
Cut holey potatoes into 2-in. chunks and drop into a large pot of well-salted water. Cut remaining potatoes into 1-in. chunks and add to pot, with enough water to cover. Bring potatoes to a boil, covered. Reduce heat and simmer until tender, 15 to 20 minutes.
Heat cream just until steaming. Drain potatoes; set pot back on stove. Pour half of hot cream into pot and add 1/2 cup butter. In batches, pass hot potatoes through a potato ricer into pot, alternating with rest of cream and 1 1/2 cups butter. Stir potatoes until smooth; then stir in 1 tsp. salt and 3/4 tsp. pepper. Add more salt or pepper if you like. Keep warm over very low heat until serving or, if making ahead, cover surface of potatoes with plastic wrap and pour a little cream on top of the wrap to keep a skin from forming. Chill.
Pat potato balls dry in a kitchen towel. Heat 2 large cast-iron or other heavy-bottomed pans over medium-high heat. Add 2 tbsp. clarified butter to each and swirl to coat. Add potato balls in a single layer and season lightly with salt and pepper. Brown well all over, 10 to 15 minutes.
Top potatoes with thyme sprigs and bake in pans or on a rimmed baking sheet in a single layer until very crisp, turning occasionally, 15 to 30 minutes. Toss with remaining 2 tbsp. butter, the parsley, and salt and pepper to taste.
Spoon warm potato purée onto a platter and mound potatoes noisette on top.