Preheat the oven to 375°F. Line a 9-inch cake pan with parchment paper.
To make the batter: In the bowl of a stand mixer fitted with the paddle attachment, add the butter, cane sugar, and peppermint extract. Mix on low until combined and there are no butter chunks.
Add the yogurt, heavy cream, and eggs and mix on low until combined.
In a separate bowl, add the all purpose flour, cocoa, baking powder, and sea salt and whisk together. Add into the bowl with the butter.
Mix on low until combined into a smooth batter, scraping down the sides of the bowl as needed.
Pour the batter into the prepared cake pan and spread out evenly. Bake for 20 minutes. Set a timer. While the cake is baking, prepare the topping.
To make the topping: In a medium bowl, add the melted butter, flour, cane sugar and light brown sugar, to a medium bowl. Fold together with a spatula until you have a paste.
Once the timer goes off, remove the cake from the oven and using your hands, crumble the topping onto the partially baked cake batter. Return the cake to the oven and bake for 15 to 20 minutes, or until a toothpick inserted in the center comes out clean. Allow the cake to cool completely before removing from the pan.
To make the glaze: Add the powdered sugar, cocoa, espresso powder, and milk into a small bowl and whisk together. Drizzle the glaze on top of the cake.