Mix flour, sugar, and salt in a stand-mixer bowl. Warm 8 tablespoons of milk to 105–110°F, then whisk in yeast, cooled melted butter, egg, and 1 teaspoon vanilla.
Pour the wet mixture into the flour and mix with a dough hook on medium for 10 minutes; the dough should remain soft and sticky.
Oil your hands and a large bowl. Shape the dough into a ball, place it in the bowl, cover, and let rise until doubled, about 1–2 hours.
Press down the risen dough and roll or press it into a 6-by-14-inch rectangle. Sprinkle cinnamon evenly over the surface.
Roll from the long side, seal the seam with a little water, and cut into fourteen 1/2-inch to 1-inch slices. Arrange on parchment, cover, and let rise 30 minutes.
For baked buns, heat the oven to 350°F and bake 15–18 minutes, until lightly golden. For fried buns, heat the optional oil to 350°F and fry in batches until golden and fluffy.
Whisk melted butter, powdered sugar, remaining milk, and remaining vanilla into a smooth glaze. Dip warm buns into the glaze and let set briefly before serving.