In a separate bowl, whisk yolks. Whisking constantly, slowly whisk about a third of the hot cream mixture into the yolks, then whisk the yolk mixture back into the saucepan with the cream. Return saucepan to medium-low heat and gently cook until mixture is thick enough to coat the back of a spoon.

Strain through a fine-mesh sieve into a bowl. Cool mixture to room temperature. Add goat cheese, fresh lemon juice, and vanilla extract and blend in a blender until smooth.

Cover and chill at least 4 hours or overnight.

Churn in an ice cream machine according to manufacturer's instructions. When the ice cream is frozen to a soft-serve consistency, use a spatula to swirl in the blueberry compote.

Remove ice cream from ice cream machine, cover tightly, and place in the coldest part of your freezer until it is firm (at least 8 hours).