In a stainless steel saucepan, bring cream and milk to a simmer. Remove from heat, stir in mint leaves, cover, and let steep for 2 hours.

In another stainless steel saucepan, whisk together egg yolks and sugar until well combined. Quickly strain dairy into pot with egg yolks, pressing on the mint with the back of a spoon to extract as much mint flavor as possible. Whisk until fully combined, then set over medium heat and cook, whisking frequently, until a custard forms on a spoon and a finger swiped across the back leaves a clean line, or until custard temperature reaches 170°F (77°C). Add salt to taste.

Pour custard through a fine-mesh strainer into an airtight container and chill in an ice bath or refrigerator until temperature drops to 40°F (4°C), about 4 hours for ice bath, up to overnight for refrigerator.

Churn ice cream according to manufacturer's instructions. While ice cream churns, melt chocolate in a microwave or double boiler and stir to combine with oil. During final minutes of churning (ice cream should have the consistency of soft serve), slowly drizzle chocolate into ice cream to form thin, lacy lines, pausing to break up larger chocolate chunks with a spoon, if needed. Transfer churned ice cream to an airtight container and harden in freezer for at least 4 hours before serving.