Brown the butter in a light-colored small saucepan over medium heat for about 5–8 minutes, stirring and scraping the bottom as it foams. When the milk solids are golden brown and smell nutty, immediately pour everything, including the browned bits, into a heatproof liquid measuring cup. Black specks and an acrid smell mean it has burned; start again rather than using bitter butter.
Cool the butter until lukewarm, about 20 minutes; do not add water while it is hot. Stir in enough room-temperature water to bring the total liquid back to ½ cup, replacing moisture lost during browning. Stir thoroughly to distribute the browned bits, then reserve 2 tablespoons of this mixture in a small covered bowl for the finish. The remaining 6 tablespoons go into the dough.
Combine the milk at 105–110°F, active dry yeast, and ¼ cup honey in a large mixing bowl. Let stand for 5–10 minutes until noticeably foamy. If there is no activity, check the yeast date and milk temperature and restart with fresh yeast. Whisk in the egg and the 6 tablespoons cooled brown butter mixture.
Add the 4 cups flour and salt. Mix to form a shaggy dough. Knead with a dough hook on medium-low for about 8–10 minutes, or by hand for 10–12 minutes, until smooth, elastic, and slightly tacky. It should pull away from most of the bowl while sticking a little at the bottom. Add extra flour 1 tablespoon at a time only if it remains a wet paste; use no more than the listed 2 tablespoons before reassessing. Avoid making a dry, stiff dough.
Form a ball, place in the mixing bowl, and cover. Let rise in a warm, draft-free place until doubled, about 60–90 minutes. Watch the size rather than the clock. While it rises, line a 9-by-13-inch metal baking pan with parchment.
Gently press the air from the dough and divide into 12 equal pieces. Cup each piece and pull its edges underneath to make a smooth, tight top, pinching the seam closed. Place seam-side down in the pan in 3 rows of 4. Cover and let rise until very puffy, nearly touching, and a gentle fingertip dent slowly springs back, about 30–45 minutes.
During the final rise, heat the oven to 350°F with a rack in the middle. Uncover and bake for 22–27 minutes, until golden brown and a center roll registers about 190–195°F inside. Loosely tent with foil if the tops brown before the centers finish.
Warm the reserved 2 tablespoons brown butter mixture just enough to liquefy it, then stir in the remaining 2 tablespoons honey. Brush all of the finish over the warm rolls. Let the rolls stand in the pan for 10 minutes before pulling apart and serving. Makes 12 rolls.