Make Buns: Microwave milk for 30 seconds or heat on the stove in a small saucepan until warm but not hot to the touch. Stir in yeast. Let stand 5 minutes. It should be frothy.
In the bowl of a food processor fitted with a dough hook, combine flour, sugar, and salt and stir with a wooden spoon to combine. Add eggs and milk mixture and stir to combine. Add butter and start mixer on low. Gradually increase speed to medium high and knead until elastic and smooth, about 10 minutes. Lightly grease a large bowl with butter. Transfer dough to bowl, cover with a kitchen towel, and let sit in a warm place until doubled in volume, about 1 hour.
Punch down dough and knead for one minute. Reshape ball and cover with a kitchen towel, let sit for 30 minutes. Butter an 8- by 11-inch baking dish (see note above).
In a small bowl, whisk together sugar, cinnamon, and nutmeg for the filling. On a lightly floured, clean work surface, roll dough out into a 16- by 8-inch rectangle. Brush with melted butter. Sprinkle surface with cinnamon sugar, leaving a 1/2-inch border on all sides. Roll dough up, jelly roll style, and press to seal edges. Use a sharp knife to cut dough into 12 pieces. Place buns, cut side down, in prepared baking dish. They should be fit together as snugly as possible without crushing them. Cover with a baking dish and let rise until almost doubled in volume, about 40 minutes.
While dough rises, adjust oven rack to middle position and preheat oven to 375°F. Bake buns until golden and puffed, about 35 minutes.
Meanwhile, Make Sauce : In a medium saucepan, melt butter over medium low heat. Add heavy cream, maple syrup, and salt, and whisk quickly to combine. Increase heat to medium high and bring to a boil. Boil hard for one minute. Brush each warm cinnamon bun with sauce.