Warm 1 cup whole milk to about 100°F.
In a measuring cup or small bowl, stir together the warmed milk, 2 tablespoons honey, and 2 ½ teaspoons active dry yeast. Let sit until foamy, 10 to 15 minutes.
In the bowl of an electric mixer fitted with the dough hook, add 4 cups all-purpose flour, 2 tablespoons sugar, and 1 teaspoon kosher salt. Add the yeast mixture and 3 large eggs.
Mix on low until the ingredients come together, then continue to mix and knead with the dough hook until the dough is soft and sticky, about 5 minutes.
Add the ½ cup unsalted butter (softened and cut into pieces). Continue to mix and knead until the butter is fully incorporated and the dough is soft and smooth, about 5 to 10 more minutes.
Lightly grease a large bowl with some of the extra butter. Transfer the dough to the greased bowl, turn to coat, cover, and let rise in a warm, draft-free place until doubled in size, about 1 hour.
Punch the dough down and divide it into 16 equal pieces. Shape each piece into a tight roll, pinching the bottom to seal.
Rub extra butter all over two oven-safe skillets or one 9x13-inch baking dish. Arrange the 16 rolls in the pan(s), dividing them evenly. Cover and let rise again until nearly doubled, about 1 hour.
Preheat the oven to 350°F. In a small bowl, whisk together 1 large egg and 1 teaspoon water to make the egg wash.
Brush the tops of the risen rolls with the egg wash. Bake for 20 to 25 minutes, until golden brown and cooked through.
As soon as the rolls come out of the oven, rub the tops with extra butter and sprinkle with flaky salt.
To make the salted maple butter: in a small bowl, beat ½ cup salted butter with 2 to 3 tablespoons maple syrup until combined and smooth. Sprinkle a little flaky salt on top.
Serve the warm rolls with the salted maple butter.