First, make the sheet cake. Position a rack in the center of the oven and preheat the oven to 350°F. Lightly spray a 9 x 5-inch loaf pan with cooking spray and line with parchment paper, leaving a 2-inch overhang on the pan's two long sides. Spray the parchment, too.
In a medium bowl, whisk together the flour, baking powder, and salt. In a liquid measuring cup, whisk together the milk and vanilla. In the bowl of a stand mixer fitted with the paddle attachment, combined the sugar and butter. Beat on medium-high speed until light, fluffy, and doubled in volume, 2 to 3 minutes, using a rubber spatula to scrape down the sides of the bowl as necessary. Reduce the mixer to low and add the egg; increase the mixer to medium-high and beat until light and fluffy, another 1 to 2 minutes. With the mixer on low, slowly pour in the oil and beat until combined.
With the mixer in low, gradually add the dry ingredients in three parts, alternating with the wet ingredients, beginning and ending with the dry ingredients. Beat until the last of the dry ingredients are just combined. Scrape down the bottom and sides of the bowl once more. Sprinkle the sprinkles over the batter and gently mix by hand with the rubber spatula until the sprinkles are evenly distributed throughout.
Pour the batter into the prepared pan and use an offset spatula to smooth its top.
Bake the cake. Bake for 22 to 24 minutes, or until the top of the cake bounces back when gently pressed. A skewer inserted into the cake should come out with a few crumbs attached. Cool completely in the pan on a wire rack before frosting. Next, make the vanilla buttercream frosting. In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and vanilla on medium-low until soft and creamy, 2 to 3 minutes. Scrape down the bottom and sides of the bowl with a rubber spatula. With the mixer on low, gradually add the confectioners' sugar and beat until combined. Scrape down the bottom and sides of the bowl. With the mixer on low, add the milk and beat until just combined. Scrape down the bottom and sides of the bowl once more, then beat on medium-high until the frosting is creamy and smooth, 2 to 3 minutes. Assemble and decorate the cake. Use an offset spatula to cover the top of the cake with the frosting completely. Garnish with sprinkles.
Serve and store. Serve immediately after decorating. The cake will keep in an airtight container for 1 day. After that, transfer to the refrigerator and refrigerate for up to 2 more days.