Melt 4 tablespoons of the butter in a microwave-safe bowl and set aside to cool to room temperature.
In a medium bowl, whisk together the yeast, 1 tablespoon honey, and warm water (about 110°F). Let sit 5 minutes until foamy; if it does not foam, the yeast may be inactive.
To the cooled melted butter, add the remaining 2 tablespoons honey and the eggs; whisk until smooth.
In a large bowl, whisk together the flour and salt. Make a well in the center and pour in the butter-egg mixture and the yeast mixture. Stir with a sturdy spatula until a thick, shaggy dough forms with no dry streaks.
Cover the bowl with plastic wrap. For same-day rolls, let the dough rise in a draft-free place (about 70°F) for 90–120 minutes until nearly tripled. For overnight rolls, let rise 30 minutes, refrigerate 8–12 hours, then bring to room temperature and wait until tripled (about 30 minutes).
Generously spray a 9×13-inch pan with cooking spray. Flour a work surface and your hands. Turn the dough onto the floured surface, shape it into a rectangular log, and cut into 12 equal pieces (cut into 4, then each into 3).
Dust pieces with flour as needed. Shape each piece into a smooth ball by pulling the dough toward the bottom and pinching underneath, then roll gently to smooth. Place the balls pinched-side down into the prepared pan in 4 rows of 3. Cover with lightly greased plastic wrap.
Let the rolls rise for 20–30 minutes (up to 45 minutes) until nearly doubled in size.
Preheat the oven to 350°F. Remove the plastic wrap and bake the rolls for 45–50 minutes until golden brown and hollow-sounding when tapped.
Brush the tops with the remaining 1 tablespoon of butter and let the rolls cool slightly before serving.