Preheat oven to 350ºF. Line a 9x13 inch sheet pan with parchment paper. If you don't have a sheet pan, use a casserole dish.
Melt 2 tablespoons of butter and spread it over the parchment paper in the pan. Sprinkle the brown sugar over the melted butter and spread out evenly. Slice the oranges into 1/4 inch thick circles. Then trim off the orange peel from the edges. Arrange the orange circles in the parchment lined pan. They should cover most of the bottom, but it's ok if there are some gaps. Set aside.
In a large bowl, whisk together the flour, granulated sugar, baking soda, baking powder, and salt. Set aside.
In a medium saucepan, add the espresso powder, cocoa powder, water, and remaining 8 tablespoon of butter. Heat over medium-low heat while whisking. Keep whisking until the cocoa powder is completely dissolved and the butter is melted. The mixture should be smooth and glossy. Remove from heat once bubbles start forming on the sides.
In a medium bowl, whisk together the greek yogurt, eggs, vanilla extract, and vinegar. Then pour the mixture into the dry ingredients. Then add the chocolate mixture from the previous step. Fold everything together using a spatula until just combined. Then add the orange zest and chopped nuts and stir until incorporated.
Pour the cake batter over the oranges in the prepared pan. Smooth into an even layer and bake for 30-34 minutes or until an inserted toothpick comes out without crumbs. Remove from oven and let cool in the pan for 5 minutes. Then cover the pan with a serving plate or platter and quickly flip it over so that the orange slices are on top. Peel off the parchment paper. Let cool for at least 10 more minutes. Optionally dust with powdered sugar before serving.