Preheat oven to 350°F. Line baking sheets with parchment paper or nonstick baking mats.
In a large bowl, whisk together the French vanilla cake mix and 1 teaspoon baking powder until uniform.
Using an electric mixer, beat in the 2 large lightly beaten eggs one at a time. Add 1/3 cup vegetable oil and 1 teaspoon pure vanilla extract; mix until well combined, scraping the bowl as needed.
Stir in 1/4 cup sprinkles or nonpareils until evenly distributed through the dough.
Shape the dough into 1-inch balls and place them 2 inches apart on the prepared baking sheets. Dip the bottom of a glass in flour and gently flatten each ball to about 1/4-inch thickness.
Bake in the preheated oven 7–8 minutes, until the edges are set and the tops are no longer glossy. Allow cookies to cool on the baking sheet for about 5 minutes, then transfer to a wire rack to cool completely.
While the cookies cool, make the frosting: in the bowl of a stand mixer fitted with the paddle attachment (or a large bowl with an electric mixer), cream 2 sticks unsalted butter (softened) on medium-high speed for about 6 minutes, until pale and fluffy. Scrape the bowl as needed.
With the mixer on low speed, slowly add 3 1/4 cups sifted confectioners’ sugar and mix until incorporated and there are no dry streaks.
Add 1 1/2 teaspoons clear vanilla extract and 2 tablespoons heavy cream or milk; blend on low speed until the frosting is moistened. If needed, add up to 1 additional tablespoon heavy cream or milk (for a total of 3 tablespoons) to reach a spreadable consistency.
Increase mixer speed and beat the frosting until smooth and fluffy, about 5 minutes.
Pipe or spread the frosting onto the cooled cookies and immediately top with additional sprinkles or nonpareils.