Potato Rolls with Caraway Salt

Place the potato in a small saucepan and add water to cover by 1". Bring to a boil; reduce heat to medium and simmer until potato is tender, about 15 minutes. Drain, reserving 1/4 cup cooking liquid in a small microwave-safe bowl.
Meanwhile, melt 4 tablespoons butter in another small saucepan. Add milk; stir until warm, about 1 minute, and set aside. Transfer cooked potato to a large bowl.
Mash with a fork, then stir in milk mixture, 1 tablespoon sugar, and 1/2 teaspoon salt (mixture will be slightly lumpy).
Rewarm potato cooking liquid in microwave (or let cool) until an instant-read thermometer registers 105°F-110°F. Stir in remaining 1 tablespoon sugar, 1 teaspoon salt, and yeast; let stand until foamy, about 10 minutes. Add yeast mixture to potato mixture and stir to combine. Add 3 1/4 cups flour with a wooden spoon and stir until a sticky dough forms.
Turn dough onto a floured surface and knead, dusting surface, dough, and your hands often with remaining 1/2 cup flour to keep dough from sticking, until dough is smooth, elastic, and slightly sticky, 7-8 minutes. Form into a ball and transfer to a buttered bowl. Turn to coat dough with butter and cover bowl with plastic wrap. Let dough rise, chilled, overnight.
Butter baking pan. Mix caraway seeds and pretzel salt in a small bowl; set aside. Punch down dough (do not knead). Cut dough in half, then roll each piece into a 12"-long log. Cut each log into 12 equal pieces, then roll each piece into a ball (for 24 total). Arrange in pan, evenly spaced, in 6 rows of 4. Cover loosely with plastic wrap. Let rise in a warm, draft-free place until almost doubled in size, about 1 hour.
Arrange a rack in middle of oven and preheat to 375°F. Whisk egg and cream in a small bowl; brush tops of rolls with some of egg wash. Sprinkle with caraway salt. Bake, rotating pan once, until rolls are cooked through and deep golden, 25-30 minutes. Transfer to wire rack and let cool in pan for 5 minutes, then run a sharp knife around edges to loosen from pan; do not separate rolls. Transfer to rack and let cool for 10 minutes. DO AHEAD: Can be made 8 hours ahead. Rewarm, wrapped in foil, in a 300°F oven until heated through. Serve warm.