Start by lining a 9x13 inch baking pan with parchment paper, leaving a bit of an overhang for easy removal later. Dust the parchment paper with powdered sugar to prevent sticking.
In a small bowl, combine 1/3 cup of cold water and the two packets of unflavored gelatin. Allow it to sit for about 5 minutes, letting the gelatin bloom and absorb the water.
In a medium saucepan, combine the granulated sugar, corn syrup, remaining 1/4 cup of cold water, and a pinch of salt. Stir over medium heat until the sugar has fully dissolved. Use a candy thermometer to monitor the temperature; you want it to reach 240°F (soft ball stage).
Once the sugar mixture reaches 240°F, remove it from the heat. Stir in the instant espresso powder until fully dissolved. Then, add the bloomed gelatin and stir until completely melted and combined.
Pour the hot mixture into the bowl of a stand mixer. Add the vanilla extract and whip on high speed for about 10 minutes, or until the mixture becomes thick, fluffy, and forms stiff peaks.
Using a rubber spatula, spread the marshmallow mixture into your prepared pan. Smooth the top with the spatula and dust with more powdered sugar.
Allow the marshmallows to set at room temperature for at least 4 hours, or overnight for best results.
Once set, lift the marshmallows out of the pan using the parchment overhang. Dust a clean cutting board with powdered sugar and use a sharp knife or pizza cutter to cut them into squares. Toss each piece in powdered sugar to prevent sticking.