Line an 8x8 or 9x9 square baking pan with parchment paper so the sides and corners are covered. Dust the parchment generously with powdered sugar, making sure the bottom and sides are coated; set the pan aside.
In the bowl of a stand mixer, dissolve the instant espresso powder in 1/3 cup cold water. Sprinkle the 2 packets of unflavored gelatin evenly over the espresso water and let sit for at least 5 minutes to soften (bloom).
In a heavy saucepan, combine the remaining 1/4 cup cold water, 1 1/2 cups granulated sugar, 1/2 cup corn syrup, and a pinch of salt. Stir over medium heat until the sugar dissolves.
Bring the sugar mixture to a boil. Cover the pan and let it steam for 2 minutes, then remove the cover and continue boiling, without stirring, until the mixture reaches 238°F on a candy thermometer (about 7–8 minutes after removing the cover).
Fit the stand mixer with the whisk attachment and start the mixer on low speed. Very carefully and slowly pour the hot sugar syrup down the inside of the mixer bowl into the gelatin mixture, taking care to avoid splattering.
Increase the mixer speed to medium-high and whip the mixture for 12–15 minutes, or until it becomes thick, glossy, and lukewarm to the touch. Add 1 teaspoon vanilla and beat briefly to incorporate.
Immediately pour or scrape the marshmallow mixture into the prepared pan (the mixture is very sticky). Smooth the top as needed (dampen a spatula with powdered sugar if it sticks).
Let the marshmallows sit uncovered at room temperature for at least 3 hours, or until fully set.
Lift the set marshmallow slab from the pan using the parchment. Dust a knife or bench scraper with powdered sugar, cut the marshmallows into squares, and dust all cut surfaces liberally with additional powdered sugar so pieces don’t stick together.
Store the marshmallows in an airtight container for up to 5 days.