In the bowl of a stand mixer combine the flour, yeast, parmesan, and salt. Add the warm milk, honey, 2 eggs, and 2 tablespoons butter. Using the dough hook, mix until the flour is completely incorporated, about 4-5 minutes. If the dough seems sticky, add the remaining 1/4 cup of flour. Cover the bowl with plastic wrap and let sit at room temperature for 30 minutes to 1 hour. Meanwhile, make the sage butter. Heat 2 tablespoons butter, the garlic, and sage in a small skillet over medium heat and cook the butter until it begins to brown and the sage is crisp, about 3-4 minutes. Let cool, then mash the garlic with a fork. Chop the crispy sage. Stir the browned butter, mashed garlic, and sage together with 4 tablespoons butter.Butter a 9x5-inch loaf. Preheat the oven to 450 degrees F. When the dough has doubled in size, turn it out onto a lightly floured work surface, punch it down, and roll it into a ball using your hands. Roll the dough into a rectangle (approximately 12 x 18 inches). Spread the sage butter over the dough. Starting with the edge of dough closest to you, roll the dough into a log.Cut the log into 8 rolls, pinch the ends to seal, then arrange in the prepared pan. Cover and let rise 15 minutes to 20 minutes in a warm place. Brush the top of the loaf with beaten egg. Bake 20 minutes, until browned on top. Reduce the oven temp to 350 degrees F, then continue baking another 25-30 minutes. Serve warm with additional butter.