In the bowl of a stand mixer, combine the flour, instant yeast, and granulated sugar. Stir with a spoon to blend, then add the salt and mix again.
Add the eggs, warm milk, vanilla extract, and melted (but cooled) butter. Using the dough hook, start mixing on low speed until the dough comes together.
Increase to medium speed and knead for 8–10 minutes, or until the dough is soft, smooth, and slightly elastic.
Cover the bowl with plastic wrap or a clean towel and let the dough rise in a warm spot for 1–2 hours, or until it doubles (or even triples) in size.
Punch down the risen dough and transfer it to a floured work surface. Roll into a large rectangle about ¼ inch thick, making sure the bottom doesn’t stick to the surface.
Spread an even layer of very soft butter all over the dough, then sprinkle the brown sugar evenly on top. Use your hands to spread it out, pressing lightly so it sticks. Sprinkle the cinnamon evenly over the sugar.
Starting at the long side, roll the dough tightly into a log and pinch the seam closed. Use a serrated knife or unflavored dental floss to cut into 12 even rolls.
Arrange the rolls in a greased 9x13-inch pan, cover, and let them rise again until doubled in size, about 30–45 minutes.
Preheat the oven to 180°C (350°F). Bake the rolls for 20–25 minutes, or until the tops are golden brown but the centers are still soft. Be careful not to overbake—this keeps them gooey and tender.