In a medium saucepan, combine milk and 12 sprigs mint. Bring to a gentle boil, cover, and remove pan from heat. Let steep 30 minutes. Strain mixture; reserve milk, and discard solids.
Combine egg yolks and sugar in the bowl of an electric mixer fitted with paddle attachment. Beat at medium-high speed until very thick and pale yellow, 3 to 5 minutes. Return milk to a simmer.
Add half of the warm milk to egg-yolk mixture; whisk until blended. Return new mixture to saucepan with remaining milk. Cook over low heat, stirring constantly, until mixture is thick enough to coat the back of a wooden spoon.
Have ready an ice-water bath. Remove saucepan from heat; immediately stir in cream. Pass mixture through a sieve set over a medium bowl. Place bowl in ice-water bath; chill. Stir in peppermint.
Freeze in ice-cream maker according to instructions. Finely chop remaining 2 sprigs mint. Add mint and chocolate; mix in machine until combined, about 30 seconds. Store in an airtight container.
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