For dough: In a large bowl, whisk together 1 1⁄2 cups flour, granulated sugar, yeast, salt, and nutmeg.
In a small saucepan, cook eggnog, butter, and 1⁄4 cup water over medium-low heat until an instant-read thermometer registers to 120° to 130°. (Butter does not need to melt completely.) Add eggnog mixture to flour mixture, and beat with a mixer at medium speed for 2 minutes. Add egg; beat at high speed for 2 minutes. With mixer on low speed, gradually beat in enough remaining 2 to 2 1⁄4 cups flour to form a soft dough. Turn out dough onto a lightly floured surface, and knead until dough is smooth and elastic, 6 to 8 minutes.
Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover with plastic wrap, and let rise in a warm, draft-free place (75°) until doubled in size, about 1 hour.
Spray a 10-inch cast-iron skillet with baking spray with flour.
Punch down dough. Turn out dough onto a lightly floured surface; roll dough into a 15x10-inch rectangle.
For filling: In a small bowl, whisk together brown sugar and cinnamon. Brush melted butter onto dough, leaving a 1⁄2-inch border on one long side. Sprinkle sugar mixture onto butter. Starting with long side opposite border, tightly roll up dough into a log, pinching seam to seal. Using a serrated knife, cut log into 10 (1 1⁄2-inch-thick) slices. Place slices, cut side up, in prepared skillet. Cover and let rise in a warm, draft-free place (75°) until doubled in size, about 45 minutes.
Preheat oven to 350°.
Bake until golden brown, 25 to 30 minutes. Let cool in skillet on a wire rack for 30 minutes.
For frosting: In a medium bowl, beat eggnog and butter with a mixer at medium speed until combined. Gradually add confectioners’ sugar, beating at low speed until smooth and creamy and stopping to scrape sides of bowl. Add 1 to 2 teaspoons water, 1 teaspoon at a time, if needed to reach a spreadable consistency. Spread onto warm rolls. Garnish with nutmeg, if desired. Serve warm.
Recipe by Paula Deen Magazine at https://pauladeenmagazine.com/eggnog-sweet-rolls/