Set up your grill for indirect heat. If using a charcoal grill, build the coals on one side only. If using a gas grill, heat one side only.
For the dough: Heat the milk in a small saucepot over the indirect-heat side of the grill to take the chill off. The milk should be between 110 and 115 degrees F. To the bowl of a mixer fitted with the dough hook attachment, add the warmed milk and yeast. Sprinkle 1 teaspoon of the sugar on top, then let stand for 10 minutes, allowing the yeast to foam.
While you’re waiting, melt the butter in a small saucepot over the indirect-heat side of the grill. When the butter has fully melted, continue to cook until browned, nutty and fragrant. Set aside to cool slightly. When the yeast mixture looks very foamy, add the browned butter and egg yolk and mix.
In a separate mixing bowl, mix together the flour, salt and remaining 2 tablespoons sugar, then add to the yeast mixture. Mix until the dough begins to climb the hook and pull away from the bowl. Continue to mix until the dough is shiny and elastic, about 5 minutes. Turn out into a greased bowl, flipping once so the top is coated as well. Cover and let rise in a draft-free place until doubled in size, about 1 hour.
For the filling: In the meantime, make the filling and prepare your pan. Combine the Parmesan, scallions, parsley, garlic and 4 tablespoons of the butter in a bowl and season liberally with salt and pepper.
Place a 10-inch cast-iron pan over indirect heat and melt the remaining 2 tablespoons of butter, swirling to coat the bottom of the pan. Set aside.
On a lightly floured surface, roll out the dough to a 5-by-15-inch rectangle. Spread the filling mixture over the entire surface, leaving a 1/2-inch clean edge on the side furthest away from you. Begin to tightly roll up into a log, pinching at the seam. Using non-flavored dental floss, slice the log into 6 pieces by first cutting in half and then cutting each half into thirds. This will ensure even sizing. Place the 6 rolls into the prepared pan, leaving some spacing in between. Cover and put in a draft-free place until almost doubled in size, about 30 minutes.
Place the pan on the indirect-heat side of the grill and cover the grill with a lid. Bake until golden brown, 18 to 20 minutes. Remove from the grill and invert onto a plate. Drizzle with hot honey and sprinkle with flakey sea salt. Serve.
To bake in the oven, bake at 400 degrees F for 18 to 20 minutes.