Combine berries, lemon zest, sugar, and salt in a heavy saucepan over medium heat. Cook, stirring occasionally, until juices leave the berries and begin to simmer, about 4 minutes, then reduce heat to low. Continue to cook, stirring occasionally, until juices thicken slightly and berries are very tender, about 7 minutes.
Transfer to a blender or food processor (or use an immersion blender in saucepan) and purée until smooth. Strain through a fine-mesh strainer. You should have about 2 cups of strained strawberry purée.
Transfer purée to a bowl and whisk in yogurt until well combined. If base tastes too sweet, add lemon juice to taste, starting with ¼ teaspoon, bearing in mind that sweetness and acidity will both be slightly muted when chilled. Chill in refrigerator until very cold, about 3 hours.
Churn in ice cream machine according to manufacturer's instructions. Enjoy immediately as soft serve, or firm up in freezer for 2 to 3 hours. Once set, frozen yogurt will be firm; defrost on counter for 10 to 15 minutes before serving.
