Combine heavy cream and whole milk in a medium saucepan and heat until steaming.

Whisk egg yolks and sugar in a separate bowl until pale yellow and thick.

Slowly pour 1 cup of warm cream mixture into egg yolks while whisking constantly.

Pour the egg mixture back into the saucepan and cook over medium-low heat, stirring continuously until it coats the back of a spoon.

Remove from heat and stir in instant espresso powder and vanilla extract.

Strain the mixture through a fine-mesh sieve into a clean bowl and cover with plastic wrap.

Refrigerate for at least 4 hours or overnight.

Churn the chilled mixture in an ice cream maker according to manufacturer's instructions.

Transfer to a freezer-safe container and freeze for 4 hours until firm.