Place the ice cream maker churn bowl in the freezer to freeze overnight.
In a small saucepan over medium heat, combine the water, 3 tablespoons of sugar, and the strawberries. Cook, stirring frequently to prevent burning, for 12–15 minutes, until the strawberries are completely broken down and the mixture has a jam-like consistency. You should have about 1 cup (240 ml). Transfer to a medium bowl and set aside.
In a large, high-walled skillet over medium heat, whisk together the oat milk and remaining 1½ cups sugar. Cook until the mixture reaches 170°F (77°C), whisking occasionally to help the sugar dissolve. Whisk in the salt, then remove the skillet from the heat.
Add the egg yolks to a large bowl. Whisk the yolks to break down, then slowly stream in about 1 cup (240 ml) of the oat milk mixture, whisking constantly, to gradually increase the temperature of the yolks. Continue streaming in the warm oat milk mixture, whisking to combine, until fully incorporated. If the custard begins to break, the milk was added too quickly.
Prepare an ice bath in a large bowl and place nearby.
Return the custard to the skillet and turn the heat to medium. Whisking continuously, bring the mixture up to 170°F (77°C). Remove the skillet from the heat and stir in the vanilla.
In a small bowl, mix together the cornstarch and 3-4 tablespoons of the warm custard, until no clumps are left. Return the custard and cornstarch slurry to the pot and stir to incorporate.
Immediately transfer the custard to a medium bowl and set in the ice bath, making sure no water gets into the bowl with the custard. Let cool completely. You should have about 5½ cups of custard.
Add 2 cups of custard to each of 2 medium bowls and set aside.
Add the strawberry preserves to the remaining custard, which should bring the amount up to about 2 cups.
In a small bowl, whisk together the cocoa powder and about ½ cup custard from a plain portion until well combined and smooth. Return to the bowl of plain custard, whisking until homogenous.
Place a 9 x 5-inch (19 x 13 cm) loaf pan in the freezer to chill.
Remove the ice cream maker churn bowl from the freezer and fill with the vanilla base. Churn for 20–25 minutes, until thickened. Remove the loaf pan from the freezer and line with plastic wrap. Spread the vanilla ice cream in an even layer on the bottom of the pan. Return to the freezer while you churn the strawberry custard.
Working quickly, wipe out the churn bowl and pour in the strawberry custard. Churn for 20–25 minutes, until thickened. Layer the strawberry ice cream on top of the vanilla and smooth the top with an offset spatula. Return to the freezer while you churn the chocolate custard.
At this point, you may need to return your churn-bowl to the freezer to cool down for about 1 hour.
Add the chocolate custard to the churn bowl and churn for 20–25 minutes, until thickened. Layer the chocolate ice cream on top of the strawberry, smoothing the top with an offset spatula. Freeze for at least 3 hours, until set.
Remove the loaf pan from the freezer and use the plastic wrap to lift the ice cream out of the pan.
Slice the neapolitan loaf and top with whipped cream and cherries.
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