Combine 16 tablespoons butter, 1 1/2 cups granulated sugar, and 1 tablespoon corn syrup in the bowl of a stand mixer fitted with a paddle attachment and cream together on medium-high for 2 to 3 minutes. Scrape down the sides of the bowl, add 2 eggs and 2 teaspoons fake vanilla, and beat for 7 to 8 minutes. Reduce the mixer speed to low, and add 2 1/2 cups flour, 2/3 cup milk powder, 2 teaspoons cream of tartar, 1 teaspoon baking soda, 1 1/4 teaspoons salt, and 1/4 cup rainbow sprinkles. Mix just until the dough comes together, no longer than 1 minute. Scrape down the sides of the bowl with a spatula.
Still on low speed, add 1 heaping cup Birthday Cake Crumbs (from the first recipe) and mix in for just 30 seconds, just until they are incorporated.
Using a 2 3/4-ounce cookie scoop (or a 1/3 cup measure), portion out the dough onto a parchment-lined sheet pan. Pat the tops of the cookie dough domes flat. Wrap the sheet pan tightly in plastic wrap and refrigerate for at least 1 hour. Do NOT bake your cookies from room temperature — they will not bake properly. Heat the oven to 350 (F).
Arrange the chilled dough a minimum of 4 inches apart on parchment- or Silpat-lined sheet pans. Bake for 18 minutes. The cookies will puff, crackle, and spread. After 18 minutes, they should be very lightly browned on the edges. The centers will show the beginning signs of color. Leave the cookies in the oven for an additional minute or so if the colors don't match and the cookies seem pale and doughy on the surface.
Cool the cookies completely on the sheet pans on a cooling rack.