Place the roughly chopped strawberries and salt in a food processor, or blender, and puree until smooth. Pour into a bowl and set aside.
Bring the half-and-half to a simmer over low heat.
Meanwhile, use a hand mixer, or stand mixer, to beat together the eggs with sugar until the mixture is a pale yellow, about 1 minute.
Temper the eggs by slowly pouring the hot half-and-half into the egg/sugar mixture, continue beating them as you do this. Once fully incorporated, transfer the mixture back into the saucepan over medium heat. Stirring constantly, cook the custard until thickened and coats the back of a spoon, about 3 to 5 minutes. Don't let boil, or the eggs will scramble!
Transfer the custard into a clean medium-sized bowl and stir in the strawberries, heavy cream, and vanilla extract. Cover with plastic wrap and place in the refrigerator for at least 2 hours, or up to overnight.
Remove the custard from the refrigerator and give the custard several stirs with a large spoon.
Transfer the custard to your ice cream maker and follow the instructions according to the manufacturer. The ice cream will be ready when it has thickened and expanded in the pan, usually about 40 to 50 minutes.
The ice cream can be served at this point, or transferred to a container with an air-tight lid and placed in the freezer to firm up, about 1 to 2 hours. If serving after the ice cream is frozen solid, then let it sit out for an hour before serving.