Make the dough. In a large bowl, whisk warm milk, yeast, and sugar.
Let sit 2–3 minutes until slightly foamy. Add melted butter, egg, yolk, vanilla, and salt. Stir in 3 1/4 cups flour, then knead by hand or with a mixer until smooth and slightly tacky, 6–8 minutes.
Add the remaining flour only if the dough is too sticky to handle.
First rise. Shape the dough into a ball, place in a lightly greased bowl, and cover. Let rise in a warm spot until doubled, about 60–90 minutes.
Cook the strawberry filling. In a small saucepan, combine strawberries, sugar, lemon juice, cornstarch, and cinnamon. Cook over medium heat, stirring, until juicy and thickened, 5–7 minutes.
You want a jammy consistency that still has small strawberry pieces. Cool to room temperature.
Prepare the pan. Grease a 9x13-inch baking dish or line with parchment, leaving overhang for easy removal.
Roll out the dough. On a lightly floured surface, roll the dough into a 12x18-inch rectangle. Spread the softened butter in a thin layer all over the surface.
Add the filling. Spread the cooled strawberry mixture evenly over the dough, leaving a 1/2-inch border on the long top edge.
This helps the roll seal.
Roll it up. Starting from the long bottom edge, roll the dough tightly into a log. Use your hands to keep the roll even. Pinch the seam to seal.
Slice cleanly. Use unflavored dental floss or a sharp serrated knife to slice into 12 even rolls.
The floss method prevents squishing and keeps layers distinct.
Second rise. Arrange the rolls in the prepared pan, cover loosely, and let rise until puffy, 30–45 minutes. Meanwhile, preheat the oven to 350°F (175°C).
Bake. Bake 22–26 minutes, until the tops are lightly golden and the centers no longer look doughy. If browning too fast, tent loosely with foil.
Make the glaze. Beat cream cheese and butter until smooth.
Add powdered sugar, vanilla, salt, and 2 tbsp milk. Beat until glossy and pourable, adding more milk if needed.
Glaze while warm. Spread the glaze over warm rolls so it melts into the swirls. Let cool slightly, then serve.