Preheat oven to 350°F and grease the inside of a 12-cup bundt pan with butter. Flour the inside of the pan, making sure all creases and center section have a thin coating of flour.
In a large bowl, whisk together flour, sugar, baking powder, and salt. Add eggs, oil, vanilla, cake batter flavoring, and milk. Using a mixer, beat on medium speed until all ingredients are incorporated, 1 to 2 minutes. Stir in sprinkles.
Pour batter into the prepared pan and bake for 40 to 45 minutes, or until center is set and a toothpick inserted into the center of the bundt cake comes out clean (a couple of crumbs are okay). Let cake cool in pan 15 to 20 minutes, then invert onto a wire rack to cool completely.
For the glaze:
To make the glaze, whisk together powdered sugar, vanilla bean paste, cake batter flavoring, milk, and food coloring if using. Whisk until smooth. Drizzle glaze over the cooled cake. Immediately add sprinkles. Let the glaze set for 20 to 30 minutes before serving. Store covered at room temperature.