Combine strawberries, lemon juice, and ¼ cup sugar in a saucepan; cook over medium heat, stirring occasionally, until thickened to a jam-like consistency, about 10 minutes.
Blend the cooked strawberry mixture until smooth and refrigerate until fully cooled; reserve 2 tablespoons of this puree for the glaze.
In a stand mixer bowl, mix flour, ⅓ cup sugar, salt, and yeast briefly to combine.
Add warm milk, softened butter, and egg to the dry ingredients; knead on medium speed for 5-7 minutes until the dough is smooth and elastic.
Cover the dough and let it rise in a warm place until doubled in size, about 2 hours.
Roll the dough on a lightly floured surface into a 16x12 inch rectangle.
Spread the cooled strawberry filling evenly over the dough, leaving a border along one long edge.
Roll the dough tightly from the opposite edge, then cut into 12 equal rolls.
Arrange the rolls in a greased 9x13 inch baking dish with space between each; cover and let rise until puffy, 30-60 minutes.
Preheat oven to 350°F; bake rolls for 20-25 minutes until golden brown and set in the center.
Allow rolls to cool slightly in the baking dish before preparing the glaze.
Beat cream cheese, softened butter, powdered sugar, vanilla extract, and reserved strawberry puree until smooth and creamy; add milk 1 teaspoon at a time for thinner consistency.
Generously spread the cream cheese glaze over the slightly warm rolls and serve immediately for best texture.