Beat the egg yolks, sugar, and brown sugar in a bowl with a whisk or electric mixer at medium speed until the still grainy, pale brown mixture has the texture of beaten batter, about 3 minutes; set aside.
Heat the milk and cream in a medium saucepan over medium heat just until bubbles frizzle all around the pan's inner edge; whisk in the espresso powder.
Whisk about 1/3 of the hot milk mixture into the beaten yolk mixture until smooth, then whisk this combined mixture into the remainder of the milk mixture in the pan.
Instantly decrease the heat to very low; if you have an electric stove, use a second burner just now turned on low.
Stir constantly, gently heating the custard until a few puffs of steam rise from its surface and it is thick enough to coat the back of a wooden spoon, about 5 minutes.
Strain through a fine-mesh sieve into a clean bowl to remove any bits of egg that have inadvertently scrambled, taking care not to scrape out any solids than may have cooked onto the bottom of the pan.
Stir in the vanilla and salt.
Chill in the refrigerator for at least 4 hours, or overnight.
Just before you make the gelato, place the custard and your ice cream machine's dasher, if possible, in the freezer for 10 minutes, just to shock them cold.
Freeze in your ice cream machine according to the instruction booklet.
Serve at once, or transfer to a container and store in the freezer for up to1 month; soften at room temperature for up to 10 minutes before serving.