Preheat the oven to 350 °F. Grease and flour a 12-cup metal bundt pan.

Melt the butter in the microwave in one-minute intervals at medium power, stirring after each until fully melted. Whisk in the cocoa powder and salt and set aside to cool to room temperature.

Once cool, whisk in the room temperature half-and-half, followed by the eggs, mixing completely after each addition, along with the sour cream, vanilla extract, and peppermint extract.

Add the flour, sugar, baking soda, baking powder, and espresso powder to the batter. Slowly fold in the dry ingredients, mixing just until combined. The batter will be thick.

Pour the batter into the prepared pan and bake until a tester inserted into the center of the cake comes out with a moist crumb; about 50 minutes. Allow the cake to cool in the pan for 15 minutes before inverting onto a cake rack and cooling completely.

While the cake cools, prepare the peppermint cream cheese frosting. Combine the cream cheese and butter in the bowl of a mixer fitted with the paddle attachment. Beat together until light and fluffy, about 5 minutes. Slowly add in the confectioners' sugar, continuing to mix until completely combined. Slowly pour in the half and half, peppermint extract, salt, and two or three drops of red food coloring and beat on medium-high speed for 2 to 3 minutes.

Spread the frosting over the top and partially down the sides of the bundt cake. Refrigerate to set the frosting.

Combine the chopped chocolate and half-and-half in a small, microwave-safe bowl. Microwave in 30-second increments at 50% power, stirring frequently, until the chocolate is fully melted and smooth. Add the melted chocolate mixture to a plastic sandwich bag. Snip off the corner of the bag and drizzle the chocolate over the cake. Cover with peppermint sprinkles. Refrigerate the cake until ready to serve.