Heat the milk to 95–100°F. Combine all sponge ingredients in a medium bowl and mix with your hands until a shaggy, wet mass forms. Cover with plastic wrap and set in a warm spot for about 20 minutes, until visibly bubbly.

While the sponge proofs, melt the butter in a small pot over medium heat. Add the chopped rosemary, sage, and thyme and cook, stirring frequently, until the butter is browned and smells lightly nutty. Do not let the herbs burn. Pour into a small bowl and cool to room temperature, or refrigerate to speed things along.

Scale the remaining dough ingredients directly into the bowl of a stand mixer. Add the active sponge and the cooled browned herb butter. Mix on low with the paddle attachment until combined.

Switch to the dough hook and mix on medium speed until a rough windowpane can be pulled: pinch off a small piece, gently stretch it into a square shape, and it should thin without tearing, showing some translucency in spots. The dough will not look perfectly smooth, but it should be one cohesive mass.

Scrape the dough into a well-oiled bowl and cover with plastic wrap. Proof for 30 minutes in a warm place, then fold: grab one side, pull it up and fold it over a third of the dough, rotate 180 degrees and repeat, then quarter-turn and repeat twice more. You should have a tight, compact square of dough.

Cover and allow the dough to double in a warm place, about 30 additional minutes. Grease a 9x13-inch baking dish. When the dough has doubled and feels light and airy to the touch, turn it out onto an unfloured surface.

Divide into 12 equal portions, about 99 g each. Arrange evenly in the prepared dish. Cover and proof until doubled and light, about 30 minutes. Preheat the oven to 350°F.

Brush the risen rolls with beaten egg. Bake for 20–22 minutes, until an instant-read thermometer inserted into the center of a middle roll reads 185°F. If you do not have a thermometer, a toothpick should come out with clinging crumbs but no streaks of raw dough.

Brush immediately with melted butter for glossy, golden tops. Serve warm.