Combine heavy cream, whole milk, 1/2 cup of the granulated sugar, instant espresso powder, and salt in a medium saucepan. Heat over medium heat, stirring occasionally, until the mixture is steaming and small bubbles just begin to form around the edges of the pan. Do not let it boil. Remove the saucepan from the heat.

In a separate medium bowl, whisk the large egg yolks with the remaining 1/4 cup of granulated sugar until the mixture is pale and creamy. This helps create a smooth base.

Slowly pour about 1 cup of the warm coffee-milk mixture into the egg yolk mixture, whisking constantly to gradually bring up the temperature of the yolks. Then, pour the tempered egg yolk mixture back into the saucepan with the remaining warm coffee-milk.

Return the saucepan to medium-low heat. Cook, stirring constantly with a wooden spoon or heatproof spatula, until the custard thickens enough to coat the back of the spoon. If you have a thermometer, it should reach 175-180°F (80-82°C). Be careful not to boil, as this can curdle the eggs.

Immediately strain the hot custard through a fine-mesh sieve into a clean bowl to ensure a silky-smooth texture, removing any tiny bits of cooked egg. Stir in the vanilla extract. Place the bowl into an ice bath (a larger bowl filled with ice and a little water) and stir occasionally until the custard is completely cool. Cover the bowl tightly and refrigerate for at least 4 hours, or preferably overnight, until it is very cold.

Once the custard base is thoroughly chilled, pour it into your ice cream maker bowl. Churn according to the manufacturer's instructions, typically for 20-30 minutes, until the ice cream reaches the consistency of soft-serve.

Transfer the soft ice cream to an airtight freezer-safe container. Press a piece of parchment paper or plastic wrap directly onto the surface of the ice cream to prevent ice crystals from forming. Cover tightly with the lid and freeze for at least 4 hours, or until it's firm enough to scoop, before enjoying.