Add the milk, sugar, and yeast to a large bowl. Mix to combine and set aside until the yeast has bloomed, about 5 minutes.
¾ cup milk, 3 tablespoons granulated sugar, 2½ teaspoons active dry yeast
Whisk in the egg, melted butter, and vanilla.
1 large egg, 2 tablespoons unsalted butter, 1 teaspoon pure vanilla extract
Slowly stir in the flour, mixing with a dough hook or wooden spoon. Once the dough forms into a shaggy ball, transfer it to a clean surface and knead until a smooth ball forms, about 2 minutes.
2½ cups all-purpose flour
Transfer the dough to a well-oiled bowl and set it in a warm place to rise until doubled in size, about 1½ hours.
Punch the dough down and transfer it to a lightly floured surface. Roll it to a 12x8-inch rectangle.
Spread the softened butter over the dough, making sure to spread it all the way to the edges.
4 tablespoons unsalted butter
In a small bowl, combine the brown sugar, cinnamon, and nutmeg. Sprinkle the spiced sugar mixture over the butter.
¾ cup brown sugar, 1 teaspoon ground cinnamon, ½ teaspoon ground nutmeg
Roll the dough tightly into a log (from the long side) and pinch the seam to seal it. Slice the dough into 12 (approximately 1-inch long) pieces.
Grease a 12-count muffin tin with butter or nonstick baking spray and place one dough round into each well, making sure to place it cut side up.
Set the pan in a warm place to rise until the muffins begin to rise over the top of the pan, about 30 minutes.
While the muffins rise, preheat oven to 375°F.
Bake for 15-18 minutes or until the muffins are set and begin to turn golden brown on top.
Allow the muffins to cool for 10 minutes before removing them from the pan.