Place racks in upper and lower thirds of oven and preheat to 350ºF. Line 2 baking sheets with parchment.
In a small bowl, sift flour, cocoa, baking powder and salt. In a separate small bowl, dissolve espresso powder in 1 tsp. warm water. Using an electric mixer on medium-high speed, beat butter and sugar until light and fluffy, about 2 minutes. Scrape down sides of bowl with a rubber spatula. Add egg, vanilla and espresso and mix, scraping sides of bowl when needed (mixture will look curdled). On lowest speed, mix in flour mixture until completely incorporated. Stir in all chips.
Using a 2-Tbsp. cookie scoop (or a 1/8-cup measure), portion dough and place 2 inches apart on lined baking sheets. Bake until edges are set and tops begin to crack, 10 to 12 minutes, switching sheets halfway through from top to bottom and front to back. Let cool on baking sheets for about 2 minutes; transfer cookies to wire racks to cool completely.
Line a baking sheet with parchment and place in freezer. Remove ice cream from freezer and let soften slightly. Using a 1/4-cup scoop or measuring cup, portion ice cream and place on a cookie, bottom side up. Top with another cookie, bottom side down, and gently press cookies together until ice cream reaches edges. Immediately place sandwich on baking sheet in freezer. Repeat with remaining cookies and ice cream. Freeze sandwiches until ice cream is firm, about 30 minutes. (For longer storage, wrap each sandwich in plastic wrap after they are firm.)