Open the chilled coconut milk cans and scoop the thick coconut cream from the top into a measuring cup until you have 1/2 cup; set aside. Pour the remaining thin coconut milk into a medium saucepan. Whisk in the sugar, cornstarch, and salt until completely smooth with no lumps.

Place the saucepan over medium heat and cook, whisking constantly, until the mixture begins to steam and thickens enough to coat the back of a spoon, about 4 to 5 minutes. Do not boil. Remove from heat immediately.

Strain the mixture through a fine-mesh strainer into a clean bowl to remove any cooked eggy bits. Whisk in the reserved coconut cream and vanilla extract. The mixture should look glossy and pour like heavy cream.

Cover and refrigerate until completely cold, at least 4 hours or overnight. The base must be very cold to churn properly; it should feel like chilled pudding.

Pour the cold base into your ice cream maker and churn according to manufacturer's instructions until it reaches the consistency of soft-serve, typically 20 to 25 minutes. It should hold its shape when scooped.

Transfer to a freezer-safe container, press plastic wrap directly onto the surface to prevent ice crystals, and freeze until firm enough to scoop, about 3 hours. Let sit at room temperature for 5 minutes before serving for the creamiest texture.