In a small saucepan, melt butter. Add the milk and heat until warm, about 115-120°F (any hotter and you risk killing the yeast).
Combine the milk, butter, eggs, honey, yeast, and salt in the bowl of a stand mixer. Add about half of the flour and stir with the dough hook until a rough, shaggy dough forms. Attach the dough hook to the mixer, turn to medium-low speed, and gradually add the remaining flour.
Knead until a mass of dough begins to form. Continue kneading on medium speed for about 10 minutes, or until a soft, smooth, and elastic dough ball is formed. The dough should be tacky and will stick slightly to the bottom of the bowl. If it’s unbearably sticky, add up to 3 tablespoons additional flour, 1 tablespoon at a time. Avoid adding too much additional flour, otherwise your rolls will become dense.
Lightly spray a large, clean bowl with nonstick cooking spray and place the dough inside. If the dough is sticking to your hands, use a plastic scraper to help remove it to the oiled bowl.
Cover the bowl lightly with plastic wrap. Let rise at room temperature until doubled in size, about 90 minutes to 2 hours.
Spray a 9 by 13-inch baking pan with nonstick cooking spray.
Gently deflate the dough. Divide the dough into 15 equal pieces. For extra precision, weigh the mass of dough and divide by 15. Weigh out each portion as you go.
Shape each piece into a tight ball by gathering pieces of dough into one central point and pinching that point firmly together to create surface tension. Roll each dough ball on a clean (unfloured) counter and place seam side down in the prepared pan.
Lightly cover with plastic wrap and let rise until doubled in size, about 1 hour to 90 minutes.
Preheat the oven to 375°F.
Brush the rolls gently with the beaten egg. Bake for 20 to 22 minutes, or until golden brown.
While the rolls bake, melt the butter in a small heatproof bowl in the microwave. Stir in the honey until combined.
Brush the rolls with the honey butter immediately after baking, then sprinkle with flaky salt. Serve warm. Store leftovers in an airtight container for up to 3 days.
Notes
* I use instant yeast to speed rising time. This dough is very rich, which means rising times will be longer. You can also use the same amount of active dry yeast. Just know the dough will take about 20% more time to rise. **If you live in a humid climate or use unbleached flour, your dough may be stickier. Add an additional 2 tablespoons (16 grams) of flour. Adding too much flour will make the rolls dry and dense. **Do not substitute all-purpose flour for bread flour. The rolls will take longer to knead and proof, and won’t turn out as soft and fluffy.
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