For the cheesecake bites: Toss the graham cracker crumbs with the melted butter and 1 tablespoon sugar in a medium bowl. Divide between two 1 1/2-cup bowls or ramekins; press into the bottoms and up the sides. Put the cream cheese in the medium bowl (no need to wipe clean); stir with a spatula. Add the heavy cream, lemon juice, vanilla and remaining 3 tablespoons sugar; mix until smooth. Spoon into the prepared bowls, cover and freeze until firm, about 2 hours. Remove the bowls from the freezer and run a knife around the cheesecakes; turn out onto a cutting board and roughly chop. Transfer to a plate, then cover and keep frozen until ready to use.
For the strawberry sauce: Meanwhile, toss the strawberries with the cornstarch and sugar in a medium saucepan, then add the lemon juice and 1 cup water. Bring to a simmer and cook, stirring occasionally, until the mixture is thick and shiny, about 30 minutes. (You should have about 1 2/3 cups sauce; cook slightly longer to reduce if necessary.) Transfer to a medium bowl and refrigerate until cold.
For the ice cream: Put the cream cheese, heavy cream, sour cream, sugar, vanilla, lemon zest and juice and salt in a food processor; process until very thick and fluffy, 1 1/2 to 2 minutes. Layer the ice cream mixture, strawberry sauce and cheesecake bites in a shallow 2-quart baking dish, reserving 1/4 cup of the strawberry sauce for serving. Cover with plastic wrap pressed directly against the surface; freeze until firm, at least 3 hours. Let sit at room temperature 20 minutes before scooping. Scoop into bowls and top with the remaining strawberry sauce and additional graham crackers.