How to Make It
Prepare the Ice Cream: Place egg yolks in a medium-size heatproof bowl. Heat cream, milk, sugar, and salt in a medium saucepan over medium-high, stirring occasionally, until bubbles just form around edges of saucepan, 4 to 5 minutes. Remove from heat; slowly and gradually pour mixture into yolks, 1/3 cup at a time, whisking constantly. Pour yolk-milk mixture into saucepan. Using your hands, squeeze burrata cheese over emptied bowl, discarding burrata peels. Cook yolk-milk mixture in saucepan over medium, stirring constantly with a heatproof silicone spatula or wooden spoon, until mixture just coats the back of a spoon, about 5 minutes. Immediately whisk mixture into burrata in bowl. Pour through a fine mesh strainer into a container; refrigerate until chilled, about 1 hour.
Pour chilled Ice Cream mixture into container of a 1 1/2- to 2-quart ice cream maker. Freeze according to manufacturer’s instructions. Transfer Ice Cream to a freezer; freeze until set, 2 to 3 hours.
Meanwhile, prepare the Tomatoes: Cut a small slit in each tomato. Bring a pot filled with water to a boil over high. Add tomatoes; cook until skins just begin to peel away, 15 to 30 seconds. Immediately plunge tomatoes into a bowl filled with ice water. Working with 1 tomato at a time, remove from ice water; peel and discard skins (leaving tomatoes whole).
Heat sugar and water in a medium saucepan over high, stirring constantly, until sugar dissolves and mixture just comes to a simmer, about 5 minutes. Stir in basil sprigs. Reduce heat to medium-low (you just want a few bubbles); add peeled tomatoes. Cook, undisturbed, 15 minutes, very gently so tomatoes don’t fall apart. Transfer saucepan to refrigerator; chill 3 to 4 hours.
Prepare the Balsamic Sauce: Whisk together balsamic vinegar and honey in a small bowl.
To serve, scoop Ice Cream evenly into parfait glasses. Top with Tomatoes in syrup and Balsamic Sauce; garnish with basil and mint leaves.