Heat 1/2 cup milk until lukewarm (105 degrees F to 110 degrees F). Pour into the bowl of a stand mixer fitted with the dough hook attachment, sprinkle in the sugar and yeast and let stand until foamy, about 5 minutes. Add the softened butter, about half each of the flour and remaining milk, and the fine salt; mix on medium speed until incorporated, about 1 minute. Add the egg and mix until incorporated, then add the remaining flour and milk; knead on medium-high speed until the dough is smooth and pulls away from the bowl, about 5 minutes (it will still be sticky).
Lightly brush a large bowl with vegetable oil and add the dough. Cover with plastic wrap and set aside in a warm place until doubled in size, about 2 hours.
Lightly brush two 12-cup muffin pans with some of the melted butter. Line a baking sheet with parchment paper. Punch down the dough. Tear off about 1 tablespoon of dough and roll into a ball, then put on the prepared baking sheet; repeat with the remaining dough. Dip the dough balls in the remaining melted butter, letting the excess drip off, and arrange 3 balls in each muffin cup to make a clover shape. Cover the pans loosely with plastic wrap and let rise in the refrigerator, at least 8 hours or up to 1 day. Cover and refrigerate the remaining melted butter.
Position racks in the lower and upper thirds of the oven and preheat to 375 degrees F. Remove the muffin pans from the refrigerator, uncover and let stand at room temperature, 1 hour. Sprinkle with sea salt and bake, rotating the pans halfway through, until the rolls are golden, 15 to 20 minutes. Melt the reserved butter, then brush on the hot rolls while still in the pan. Serve or let cool and freeze.
Make It Ahead: Let the rolls cool completely, then freeze in a resealable plastic bag for up to 1 week. To reheat, arrange on a baking sheet, cover with foil and bake at 375 degrees F until warmed through, 25 minutes. Brush with melted butter.