Whisk eggs and salt together in a medium bowl. Set aside.
Heat heavy cream, half-and-half, sugar, nutmeg, and cinnamon in a saucepan over medium heat, stirring occasionally until sugar is dissolved and cream just starts to bubble. Remove from heat.
Whisk 1 cup of the hot cream mixture into the eggs in a slow stream. Whisk the egg mixture into the remaining cream mixture in the saucepan. Cook over low heat, stirring constantly, until the mixture just coats the back of a spoon and reaches 175 degrees F (90 degrees C), about 5 minutes. Remove from heat; whisk in vanilla and almond extracts.
Let cool completely, stirring occasionally, about 15 minutes. Chill ice cream base, covered, 8 hours to overnight. Meanwhile, prepare pie crust squares and cherry pie filling.
Roll pie crust to 1/8-inch thickness. Cut into 2-inch squares. Mix sugar and cinnamon together in a small bowl.
Heat oil in a frying pan to 350 degrees F (175 degrees C). Fry pie crust squares until cooked through and beginning to turn golden. Remove squares from the oil with a slotted spoon or spatula, shaking lightly to discard excess oil, and transfer to bowl of cinnamon-sugar mixture. Stir to coat.
Transfer coated pie crust squares to a paper towel-lined plate to cool. Store in an airtight container until ready to churn ice cream.
Line a baking sheet with waxed paper. Spread cherry pie filling in a single layer and freeze until ready to churn ice cream.
Freeze base in an ice cream maker according to manufacturer's instructions, about 20 minutes. Add pie crust pieces and dollops of frozen pie filling, broken off with a spoon, in the last 5 minutes. Transfer to an airtight container and freeze until firm, about 4 hours.
