Preheat oven to 350º and line a 12x16x1-inch sheet pan with waxed paper.
In a large bowl, whisk together first 5 ingredients.
Whisk in the hot milk and melted butter, then the eggs and ¾ teaspoon peppermint extract. Whisk until well-combined and smooth.
Pour batter into prepared pan and spread evenly. Bake until toothpick comes out clean, about 25 minutes. Cool cake completely in pan on rack.
Make frosting. In the bowl of a stand mixer with paddle attachment, put the cream cheese and butter. Mix until well combined, scraping down sides as needed. Add the remaining ¾ teaspoon peppermint extract. With machine on low, gradually add the powdered sugar until incorporated, then increase speed and mix until well combined and fluffy.
Remove cake from pan by running knife around the sides of pan and inverting carefully onto work surface. Peel off paper carefully and then cut cake crosswise in half. Place one half on serving platter, tuck waxed paper underneath edges, and spread with 2 cups frosting. Top with other half and spread top and sides with remaining frosting. Remove paper. Serve room temp and refrigerate leftover cake.