Preheat the oven to 350 degrees F.
Combine the strawberries, balsamic vinegar, and 2 tablespoons of sugar in a roasting pan. Toss gently to coat and bake in the oven for 10 minutes until the strawberries are soft and release their juices.
Let the strawberries cool lightly for about 15 minutes.
Transfer the strawberries and juice to a blender and puree until smooth.
Strain the strawberry puree through a fine mesh strainer. You should get about 2 cups. Set aside.
In a medium heatproof bowl, whisk the yolks with 6 tablespoons of granulated sugar just to break them up. Set aside.
In a heavy saucepan over medium low heat, combine cream, milk, and salt and let it heat up until you see the liquid bubbling at the edges.
Measure out half a cup of cream mixture.
While whisking the eggs constantly, whisk the hot cream mixture into the eggs until smooth. Continue tempering the eggs by adding another ½ cup of hot cream to the bowl with the yolks.
Pour the cream-egg mixture back to the saucepan. Add the strawberry puree.
Cook the mixture over medium heat, stirring constantly until it is thickened and coats the back of a spatula, about 5-7 minutes.
Strain the base through a fine-mesh strainer into a clean container to remove any egg pieces that were cooked during the process.
Pour the mixture into a 1-gallon Ziplock freezer bag and submerge the sealed bag in an ice bath until cold, about 30 minutes.
Refrigerate the ice cream base for at least 4 hours or overnight. I like to refrigerate the base overnight for the most flavor.
Pour the ice cream base into the frozen canister of your ice cream machine and follow the manufacturer’s instructions. Spin until thick and creamy about 30-35 minutes.
Pack the ice cream into a storage container, press a sheet of parchment directly against the surface and seal with an airtight lid. Freeze in the coldest part of your freezer until firm, at least 4 hours.