Preheat the oven to 350°F. Grease and line an 8-inch round cake pan with parchment paper or lightly grease it.
In a large bowl, cream together 3/4 cup unsalted butter, softened, and 7/8 cup granulated sugar until light and fluffy.
Beat in 3 large eggs, one at a time, mixing well after each addition. To help prevent the batter from curdling, add about 2 tablespoons of flour with each egg.
Gently fold in the remaining 1 1/2 cups all-purpose flour and 1 teaspoon baking powder, then stir in 2 tablespoons milk and 1 teaspoon vanilla extract until just combined.
Pour the batter into the prepared pan and bake for 25 to 30 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool completely in the pan.
Meanwhile, make the buttercream: in a medium bowl, beat 1/2 cup unsalted butter, softened, until pale and creamy. Gradually add 1 2/3 cups powdered sugar, beating well after each addition.
Stir in 1/2 teaspoon vanilla extract, then add up to 2 tablespoons milk, one tablespoon at a time, until the buttercream is smooth and spreadable.
Crumble the cooled cake into a large bowl, breaking up any large pieces. Add two-thirds of the buttercream and mix with your hands until a dough forms. Add more buttercream if needed.
Roll the mixture into 1- to 1 1/2-inch balls and place them on a baking sheet lined with parchment paper.
Melt about 2 ounces of white chocolate. Dip the tip of each of the 22 lollipop sticks into the melted chocolate, then insert it halfway into each cake ball. Refrigerate for at least 30 minutes.
Melt the remaining 12 ounces of white chocolate in a double boiler or in the microwave in 30-second intervals, stirring until smooth.
Dip each chilled cake pop into the melted chocolate, turning to coat evenly, and tap off any excess.
Before the coating sets, sprinkle colorful sprinkles over each cake pop. Stand the pops upright to let the coating set completely, then chill for another 15 to 20 minutes.
Store the cake pops in an airtight container in the refrigerator for up to 3 to 4 days. Bring them to room temperature for 15 to 20 minutes before serving.