Make the dough. In a mixing bowl, whisk together flour, sugar, and yeast. In a saucepan, warm the milk, water, butter, and vanilla until the butter melts. Add the dry mixture, warm liquids, and egg to a stand mixer fitted with a dough hook. Mix and knead until the dough is smooth and pulls away from the sides. Mix in the sprinkles until evenly distributed.
First rise. Shape the dough into a ball and transfer to a greased bowl. Cover with plastic wrap and let rise in a warm place for 30 minutes.
Make the filling. While the dough rises, stir together softened butter, brown sugar, granulated sugar, and cinnamon in a small bowl to create a thick filling.
Roll and cut. Roll the dough into a 10x14-inch rectangle on a floured surface. Spread the filling evenly on top. Roll the dough tightly into a log, seam-side down, and slice into 12 even pieces using a serrated knife.
Second rise and bake. Arrange the rolls in a greased 9x13-inch pan. Cover and let rise for 1 hour in a warm place. Preheat oven to 350°F, then bake for 30 minutes, until golden on top.
Frost and serve. While rolls cool slightly, beat together the frosting ingredients until smooth and fluffy. Spread the frosting over warm rolls, top with extra sprinkles, and enjoy!