Place the yolks, sugar, and salt in a medium heat-safe bowl and whisk to combine.
Place the milk, cream, and espresso powder in a medium pot and cook over medium-low heat until small bubbles appear around the edges and the liquid is steaming.
In a thin, slow stream, pour the hot cream mixture into the yolk mixture while whisking.
Transfer the custard back to the pot and cook on low heat (whisking) until slightly thickened.
Remove from the heat and stir in the vanilla. Chill the mixture overnight in the refrigerator.
Place the chilled ice cream base in the prepared ice cream maker, and churn according to the manufacturer's directions.
Layer the ice cream in a loaf pan, sprinkling in the donuts as you go along.