In the bowl of a stand mixer, combine the warmed milk with 1 tablespoon of the sugar. Sprinkle the yeast over the mixture and allow it to stand for 1 minute. Whisk briefly, then let it rest an additional 5 minutes until the yeast appears foamy. Incorporate the 4 tablespoons of melted butter along with the remaining sugar and the fine sea salt.

Gradually add the flour, starting with half-cup increments, mixing well after each addition. Once the mixture becomes too thick to whisk, transition to the dough hook attachment and continue mixing on speed 2, or proceed by hand with a firm spatula. Incorporate up to 4 cups of flour, holding back the final tablespoons unless necessary. The dough should be slightly sticky but not adhere to dry fingertips.

Knead using the dough hook for 2 minutes, or by hand for 10 minutes, until smooth and elastic. Transfer the dough to a greased bowl, cover with plastic wrap, and set in a warm area to rise for approximately 2 hours or until doubled in volume.

Turn the risen dough out onto a floured work surface and divide it into 15 to 24 equal sections, depending on the dimensions of your baking dish. Form each portion into a ball and arrange them in a greased 9×13-inch casserole dish or baking sheet.

Loosely cover the rolls with plastic wrap and allow them to rise for another 30 to 60 minutes in a warm environment until visibly puffed, avoiding over-proofing. Bake in a preheated 375°F oven for 23 to 25 minutes, or until the tops achieve a golden-brown hue.

Upon removal from the oven, immediately brush the tops with melted butter and allow the rolls to cool to room temperature before serving.