Make the strawberry syrup first. Wash the strawberries and remove the green tops. Dice the strawberries into ¼-inch pieces. Reserve half of the chopped strawberries to fold into the ice cream later.
Add the other half of the diced strawberries to a sauce pot with the sugar. Stir and cook over medium heat until the strawberries release their liquid and form a syrup in the saucepot.
Let the strawberries simmer in the syrup, stirring occasionally until they break down and lose their shape and the syrup has thickened slightly. You'll know it's ready when you drag a wooden spoon across the bottom center of the pot and the syrup stays parted for a few seconds. Remove the syrup from the heat and stir in the vanilla and salt. Set the strawberry syrup aside to cool.
While the syrup is cooling, prepare the ice cream base. Add the sweetened condensed milk, heavy whipping cream, vanilla extract, and salt to a bowl. Use a hand mixer to whip the mixture until it is light and fluffy, and has formed soft peaks.
Fold the reserved chopped fresh strawberries into the ice cream base. Transfer the whipped ice cream base to a freezer-safe container. Drizzle the cooled strawberry syrup over the ice cream and gently swirl it into the mixture.
Cover the ice cream and freeze for eight hours before scooping and serving.