Preheat your oven to 350°F. Line a 9x9 baking pan with a parchment paper sling – two long pieces crisscrossed so the ends hang over all four sides.
In a food processor, pulse the flour, sugar, and salt together. Add the cubed cold butter and vanilla. Pulse until the mixture looks like coarse sand with a few pea-sized butter pieces remaining. If it doesn't hold together when squeezed, add the milk and pulse once or twice. Press the mixture evenly into the bottom of the prepared pan. Prick all over with a fork.
Bake the crust for 18 to 22 minutes, until the edges are lightly browned and the center looks dry and golden. Let the crust cool completely in the pan on a wire rack. Do not skip cooling.
While the crust bakes, combine the chopped strawberries, sugar, and lemon juice in a small saucepan. Cook over medium heat, stirring occasionally, until the berries break down and the liquid thickens – about 8 to 10 minutes. Let cool completely (spread on a plate in the fridge to speed it up).
In a large bowl, whip the cold heavy cream with an electric mixer until soft peaks form. In a separate bowl, whisk together the sweetened condensed milk, vanilla, and salt. Gently fold the whipped cream into the condensed milk mixture. Fold, do not stir.
Pour the ice cream base over the cooled crust and spread evenly. Drop spoonfuls of the cooled strawberry swirl over the top. Use a knife or skewer to swirl it through – just a few figure-eight motions. Do not over-swirl.
Cover the pan tightly with plastic wrap or foil. Freeze for at least 4 hours, but preferably 6 to 8 or overnight.
Use the parchment sling to lift the entire block out of the pan. Let sit at room temperature for 5 minutes. Slice with a sharp chef's knife, wiping clean between slices. Serve immediately.