In a medium-size saucepan over medium heat, cook strawberries with ½ cup sugar, stirring often, until berries begin to soften, about 3 minutes. Remove from heat; then pour half the berries into a bowl and mash. Set all berries aside.
In a medium-size saucepan, bring cream and milk to a simmer; then remove from heat. Meanwhile, in the bowl of a standing mixer or with a hand mixer, whisk egg yolks with remaining ½ cup sugar until pale and fluffy, about 3 minutes. Pour ½ cup of hot cream/milk into egg mixture, whisking as you go; then transfer egg mixture back into pan. Cook over medium heat, stirring continuously, until custard reaches 175°F on an instant-read thermometer (it will thicken noticeably).
Remove pan from heat and pour mixture through a strainer into a bowl. Stir in salt and mashed strawberries (not chopped berries). Refrigerate 6–12 hours. For faster results, stir over a bowl of ice, but let custard sit several hours for best texture.
Once mixture is done chilling, it should register between 35°F and 40°F on an instant-read thermometer. Pour into ice-cream maker, leaving 1 inch at top to allow for expansion, and prepare according to freezer instructions. Add remaining berries and juices during final 5 minutes of chilling. For firmer ice cream, chill in freezer at least 3 hours.