Heat cream and milk in a medium saucepan over medium-low heat until steaming but not simmering, 3 to 5 minutes. Add mint leaves, cover, and turn off heat. Steep for at least 1 hour, then strain through a sieve (discarding mint) and return mixture to the saucepan.

Prepare an ice bath by filling a large bowl ⅔ of the way with ice and water.

Return pot to medium-low heat, stirring occasionally, until steaming and warm but not simmering, 3 to 5 minutes.

Whisk egg yolks and sugar together in a medium bowl for about 1 minute until mixture is silky and light yellow. Whisk about ⅓ of the warm milk into the egg mixture until combined. Add egg mixture to the remaining warm milk and stir to combine. Cook, stirring constantly, for 5 to 7 minutes, until custard thickens and coats a spatula.

Strain custard through a sieve into a medium bowl. Place bowl in the ice bath for about 10 minutes, stirring to cool down custard.

Transfer cooled custard to a blender. Add tarragon leaves, cover, and pulse until combined—the mixture will be pale green. Transfer to a lidded container, cover, and refrigerate at least 4 hours or overnight.

Stir custard to distribute herbs evenly before pouring it into the chilled bowl of your ice cream maker. Process according to the manufacturer’s directions. Transfer to a lidded container, cover, and freeze for at least 2 hours or overnight. Soften at room temperature for about 5 minutes before serving.