Puree the watermelon chunks in a blender or food processor until smooth; strain the juices from any remaining solids and set aside, discarding the solids.
Heat 1 cup of heavy cream with the sugar in a saucepan over medium-low heat until warmed. Stir the mint leaves to the warmed cream. Cover the saucepan and remove from heat. Allow the mixture to steep at room temperature for 1 hour. Strain into a bowl; discard the mint leaves.
Whisk the egg yolks in a bowl until gently beaten. Slowly pour the mint-infused cream into the egg yolks, whisking constantly to prevent cooking the egg; scrape the mixture back into the saucepan and place over medium heat. Stir constantly, scraping the bottom while cooking, until the mixture thickens and coats the spatula; pour the mixture through a strainer into a bowl along with 1 cup heavy cream. Mix the watermelon juice into the cream mixture.
Pour into an ice cream maker and freeze according to manufacturer’s instructions.