Using a stand mixer* with a dough hook, place the warm milk (think baby bath water warm), yeast, and honey/sugar together. If you are using standard dry yeast, wait about 5 minutes to ensure it's activating (yeast will begin to foam). If you're using instant, you can proceed to the next step.
Turn the mixer on and add 3 cups of flour. Mix together. Add salt, softened butter and eggs. Continue to mix on low.
Slowly add the additional 2 1/2 cups of flour. In pouring your last cup, do so slowly, watching the dough. When you've added enough flour, the dough will be soft and just slightly sticky. It will barely start pulling away from the bowl. Leave the mixer on for an additional 8-10 minutes to knead.
Take your dough hook out, and cover the bowl to let the dough rest and rise for 45-60 minutes, or until doubled in size.
Grease 2 baking sheets or line with parchment paper. When the dough has doubled, place it on a floured work surface. To form, gently pinch off pieces of dough, about the size of a small tangerine. Place the shaped dough on the baking sheet, and continue to shape the remaining dough, keeping the rolls about an inch apart (they'll rise and spread). When the rolls are shaped, cover with a sack cloth towel and place in a warm, draft free place for about 30 minutes, until dough has grown slightly and is soft. Preheat oven to 350° F.
Bake for 16-20 minutes. If the tops begin to brown too quickly mid-bake, place a piece of aluminum foil over top to slow down the browning.
The rolls will be done when the bottoms are lightly golden brown. Take out of the oven and brush with butter while warm.