To make the Mozzarella Ice Cream: In a non-reactive bowl, whisk egg yolks and vanilla until well incorporated.
In the bowl of a food processor, pulse Mozarella cheese until pieces are the size of coarse salt. Transfer to a saucepan and add cream and milk. Bring to a simmer over medium heat, whisking. Add sugar and continue to whisk until sugar is dissolved.
Pour Mozzarella mixture through a fine-mesh strainer into egg yolk mixture, whisking vigorously the entire time to push through. Pour entire mixture through a clean fine-mesh strainer into a clean saucepan over medium heat and cook to 175°F. Transfer mixture to a clean bowl, cover the surface of the mixture with plastic wrap, and set aside in the refrigerator overnight.
Transfer chilled mixture into an ice cream maker and freeze according to manufacturer’s instructions.
To make the Tomato Pearls: Divide ice between 2 metal bowls. Nestle a smaller metal bowl into each bowl on top of ice. Add oil to 1 of the smaller bowls and cold water to the other. Set aside.
In a small saucepan, bring tomato juice and agar agar to a boil, gently stirring until agar agar is completely dissolved. Use a caviar maker or small disposable pipette to add drops of tomato mixture into bowl with oil (if using pipettes, you might need several— they may clog as the liquid cools). Let drops sit for 1 to 2 minutes, then transfer them to the bowl with cold water.
To make the Candied Basil Leaves: Arrange leaves on a paper towel, making sure they are completely dry. In a small bowl, whisk egg whites. Place sugar in another small bowl.
One at a time, drag leaves through egg whites, making sure leaves are completely coated and shaking off any excess, then drag through sugar, making sure leaves are completely coated. Arrange coated leaves on parchment and set aside to dry completely (may take as long as overnight, depending on humidity).
To serve: Scoop ice cream and arrange in serving dishes. Strain tomato pearls through a fine-mesh strainer and arrange on top. Garnish with basil leaves.
