Position rack in the center of the oven and preheat the oven to 375 F. Line 2 baking sheets with parchment paper or silpats.
In a small bowl or cup, mix together the instant espresso powder and water and set aside.
Sift together the flour, cocoa, baking soda, and salt into a medium bowl.
In the work bowl of a stand mixer fitted with the paddle attachment, beat together the butter and brown sugar on low speed until just combined. Increase the speed to medium and beat until fluffy and smooth, about 3 minutes. Add the egg and vanilla and beat for another 2 minutes.
Add half of the flour and half the milk and beat on low until just incorporated. Scrape down the sides of the bowl. Add the espresso-water mixture and remaining flour mixture and milk and beat until completely combined.
Using a spoon, drop about 1 tablespoon of batter onto the prepared baking sheets, spacing them at least 2 inches apart. Bake for about 10 minutes or until the cakes spring back when touched. Remove the baking sheets from the oven and let the cakes cool on the sheet for about 5 minutes before transferring them to a rack to cool completely.
Whipped Chocolate Ganache
Put the chocolate in a large, heatproof bowl. In a large, heavy-bottomed saucepan over medium heat, heat the cream until it bubbles. Pour the hot cream over the chopped chocolate and let sit for about 10 minutes, until the chocolate is melted. Add the salt and vanilla and stir with a wooden spoon or silicone spatula just to combine.
Refrigerate the ganache for at least 1 hour and up to 24 hours, until it is firm. Using a hand mixer or stand mixer, beat the ganache at medium speed until it is softened and lighter in color, about 2 minutes.
Note: Ganache is easiest to spread on whoopie pies when at room temperature.