Cream butter and sugar in a medium bowl. Beat in eggs and vanilla. Add flour, cocoa powder, espresso powder (if using) and oil. Beat until thoroughly combined.
Drop by rounded teaspoonfuls about 1 inch apart onto a preheated ungreased waffle iron. (To avoid burnt fingers, use two spoons, one to scoop and one to scrape dough onto the waffle iron.) Bake until the cookies are baked through, but not browned, 1 to 1 1/2 minutes. Waffle irons vary, so watch closely.
Transfer to a wire rack to cool until just warm. Dust cookies with confectioners' sugar while still slightly warm. (See Topping Variations.).
Topping Variations: Omit confectioners’ sugar. Drizzle cooled cookies with melted bittersweet and/or white chocolate. Or, make a peppermint drizzle: Mix 1 cup confectioners’ sugar, 4 teaspoons water and 1/4 teaspoon peppermint extract; add natural green food coloring, if desired.
To Make Ahead: Store in an airtight container for up to 1 day. Dust with additional confectioners’ sugar just before serving.