Preheat the oven to 350°F (175°C). Grease and lightly line a square baking pan with parchment paper. Pour the boiling oat milk over the quick oats, stir in the espresso powder, and let the mixture stand for about 10 minutes until the oats become soft.

In a large bowl, whisk together the melted butter, brown sugar, and granulated sugar until smooth. Add the eggs one at a time, then mix in the extra egg yolk until fully combined.

Stir the softened oat mixture into the butter mixture. Add the cocoa powder and mix until the batter develops an even chocolate color.

In another bowl, combine the all-purpose flour, oat flour, baking soda, and salt. Gradually fold the dry ingredients into the wet ingredients until no dry flour remains. Fold in 1 cup of the dark chocolate chunks.

Transfer the batter into the prepared baking pan and spread it evenly. Sprinkle the remaining chocolate chunks across the top for extra chocolate in every slice.

Bake for 32–35 minutes, or until a toothpick inserted near the center comes out with only a few moist crumbs. Remove the cake from the oven and allow it to cool completely on a wire rack.

To prepare the frosting, beat the softened butter until creamy. Mix in the cocoa powder, powdered sugar, yogurt, salt, and enough milk to create a smooth, spreadable frosting.

Spread the chocolate frosting evenly over the cooled cake. Let it rest for 15 minutes before slicing into 12 equal pieces and serving.