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Preheat oven to 350°F. Butter a 9-by-13-inch cake pan. Line pan with parchment, leaving a 2-inch overhang on long sides. Butter parchment, then dust with flour, tapping out excess.
In a medium bowl, whisk together milk, eggs, and vanilla.
In a stand mixer fitted with the paddle attachment, beat flour, sugar, baking powder, and salt on low speed until well combined, about 1 minute. Continue beating while gradually adding butter until mixture is crumbly, about 3 minutes.
Slowly add half of milk mixture; increase speed to medium and beat until fluffy, about 1 minute. Slowly add remaining half of milk mixture, scraping down bowl as needed. Beat until incorporated, about 30 seconds.
Remove bowl from mixer; fold in sprinkles with a spatula.
Pour batter into prepared pan; tap on work surface to release any air bubbles. Bake until golden brown and a tester inserted in center comes out clean, 30 to 35 minutes.
Transfer pan to a wire rack; let cool completely. Remove cake from pan using parchment. Transfer to a cake plate; remove parchment. Spread frosting evenly over top, and top with large sprinkles.