In a small bowl, sift together flour, cocoa, espresso powder, baking powder, and salt.
In a medium bowl, beat butter with a mixer at medium speed for 1 minute. Add both sugars and beat until very well blended, about 5 minutes. Add eggs and vanilla, and beat until fully incorporated. Fold flour mixture into sugar mixture with a spatula. Then fold in chocolate. Divide batter evenly among four 4-ounce ramekins and then place ramekins in a 9" x 9" pan. Cover and refrigerate 4 hours or up to 2 days.
Preheat oven to 350° F.
Let ramekins stand at room temperature 10 minutes. Uncover pan and bake for 24 to 26 minutes, or until center of cakes puff and are slightly crusty on top. If you want an extra gooey center, start checking the cakes after 20 minutes of baking time, removing the cakes from the oven before they are fully puffed in the center - take care to not over bake. A longer baking time will ensure a more baked center crumb. Both ways are fabulous!
Sprinkle with powdered sugar. Serve immediately for the most gooey, fudgy texture.
