Whisk 1/4 cup milk and cornstarch together in a small bowl until smooth to make a slurry.
Combine the remaining milk, sugar, and cocoa powder in a small saucepan over medium heat. Bring to a boil while stirring constantly. Add espresso powder and the cornstarch slurry. Bring back to a boil; stir gelato mixture until thickened like pudding, about 2 minutes. Remove from heat; transfer to a metal bowl. Stir in vanilla extract and salt.
Fill a large bowl with ice water; set the bowl of gelato mixture inside. Stir frequently until cooled, about 7 minutes. Remove metal bowl from ice bath; cover with plastic wrap. Refrigerate until completely chilled, about 20 minutes.
Transfer gelato mixture into an ice cream maker; freeze according to manufacturer's instructions, about 20 minutes. Pour gelato into a freezer-safe container; cover and freeze until solid, about 10 minutes.