Special equipment: Two 12-cup muffin tins
Warm 1/4 cup of the milk in a small saucepan over medium-low heat to anywhere between 100 and 115 degrees F. Pour the milk into the bowl of a stand mixer fitted with a hook attachment. Add the sugar and yeast, and let stand until foamy, about 5 minutes. Add the softened butter, half the flour, half the remaining milk and fine salt. Mix on medium-high speed just to incorporate. Add the egg, mix until fully incorporated, then add the remaining flour and milk. Knead with the hook on medium-high speed until the dough is smooth and pulls away from the sides, about 5 minutes (the dough will still be sticky).
Lightly butter a large bowl with some of the melted butter, and turn the dough out into it. Cover with plastic wrap, and let stand in a slightly warm place until doubled in size, about 2 hours.
Lightly butter two 12-cup muffin tins with some of the melted butter. Punch the dough down. Divide it into 24 even pieces, and roll them into rough balls. Roll each ball in the melted butter to coat, letting the excess drip off, and plop it into a muffin cup. Cover lightly with plastic wrap, and let rise in the refrigerator at least 8 hours and up to 1 day.
Position oven racks in the lower and upper thirds of the oven, and preheat to 375 degrees F. Remove the tins from the refrigerator, and let stand at room temperature 1 hour (the dough won't puff up significantly). Sprinkle evenly with sea salt, and bake, rotating the tins halfway through, until just lightly golden but still rather pale and almost cooked through, 10 to 15 minutes.
Let the parbaked rolls cool slightly in the tins, then transfer to cooling racks to cool completely. Pack them in resealable plastic bags, label and date, and freeze for up to 2 weeks.
To serve, preheat the oven to 350 degrees F. Bake the frozen rolls on a parchment-lined baking sheet until warmed through and golden brown on top, about 20 minutes. Brush the warm rolls with melted butter.
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