Position a rack in the center of the oven and preheat to 350 degrees F. Grease the bottom and sides of a 10-inch cast-iron skillet with about 1 tablespoon of the butter.
Whisk the flour, baking powder and salt together in a medium bowl. Beat the remaining butter and 1 1/2 cups sugar together in a large bowl with an electric mixer on medium-high speed until light and fluffy, about 4 minutes. Add the egg and vanilla, beating to incorporate. Add the flour mixture in 2 additions, beating on medium-low speed until incorporated.
Use a rubber spatula to scrape the dough into the prepared skillet, gently spreading it to the edges and smoothing the top. Bake until the edges are brown, and the top is golden but still slightly soft, 45 to 50 minutes. Cool on a wire rack 15 minutes.
To make the glaze, whisk the milk and 1/2 cup confectioners' sugar together in a small bowl. (If the glaze is too thin add another tablespoon confectioners' sugar.) Drizzle the glaze over the cookie and top with the rainbow sprinkles. Slice into wedges and serve warm.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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