Warm the milk to 110 to 115 degrees F. Whisk in the granulated sugar and active dry yeast. Let stand for about 10 minutes until thick and bubbly.
Add the room-temperature eggs, cooled melted butter, honey, and kosher salt. Whisk until combined. Add 3 cups of bread flour and mix. Gradually add more flour about 1/4 cup at a time until the dough pulls away from the bowl while remaining slightly sticky and moist.
Turn the dough onto a lightly floured surface and knead gently for about 4 minutes, pushing, folding, and rotating the dough until smooth and elastic. Avoid adding too much extra flour.
Grease a large clean bowl with olive or vegetable oil. Place the dough in the bowl, turning once to coat lightly. Cover tightly with plastic wrap and let rise in a warm, draft-free spot for 1 to 2 hours, until doubled in size.
Gently punch down the risen dough and move it to a clean surface. Divide it into 12 equal pieces. Shape each piece by pulling the edges downward and tucking them underneath, then pinch the seams closed to create smooth, taut dough balls.
Grease a 9x13-inch baking pan with melted butter. Place the shaped rolls seam-side down in the pan. Cover loosely with plastic wrap and let rise in a warm spot for 1 hour, until puffed and touching. Preheat the oven to 350 degrees F.
Brush the risen rolls with melted butter. Bake on the center rack for 15 to 20 minutes, until golden brown. Meanwhile, whip the softened butter and honey on high speed until pale and fluffy. Remove the rolls from the oven and immediately brush generously with the whipped honey butter. Cool slightly before serving.