Proof the yeast by combining warm milk, 1 tsp sugar, and yeast in a small bowl. Let sit 5-10 minutes until foamy.
Mix flour, salt, and remaining sugar in a stand mixer. Add yeast mixture, melted butter, eggs, and vanilla. Knead on medium speed for 5-7 minutes until smooth.
Rise the dough covered in a warm spot for 1 hour until doubled.
Cook the strawberry filling by simmering chopped strawberries, sugar, lemon juice, and cornstarch over medium heat for 8-10 minutes until thick. Cool completely.
Roll out dough to a 12x16 inch rectangle, spread cooled filling, leaving a half-inch border on one short side.
Roll tightly from the short side, seal seam, and cut into 9 rolls. Place in greased 9x13 dish.
Rise covered for 30 minutes. Preheat oven to 375°F and bake 25-30 minutes until golden.
Make icing by beating cream cheese, butter, powdered sugar, lemon zest, and milk. Drizzle over warm rolls.