Melt the butter in the microwave, then pour into a bowl with both the sugars and mix together.
Add a splash of hot water to the espresso powder and mix together to form a paste. Add to the butter/sugar mixture and stir everything together.
Add the egg and vanilla and stir until combined.
Add the dry ingredients and mix together until a soft dough forms.
Chill the dough for 30 minutes so it's easy to scoop.
Preheat oven to 350°F / 180°C and line a cookie sheet with parchment paper.
Spoon walnut-sized scoops of dough and roll them into balls in your hands. Place the cookie dough balls on the baking sheet, spaced well apart to allow for spreading.
Bake for 12 minutes, then remove from the oven and let sit on the tray for at least 10 minutes before moving to a wire rack to cool.
Notes
You may want to bake the cookies in batches if you can't fit them all on one baking sheet. I like to use two baking sheets so I can have one cooling while the next one goes into the oven.