In a large mixing bowl, stir together the warm milk and instant dry yeast and let sit for 5 minutes until slightly foamy on top.
Whisk in the egg, 1/3 cup granulated sugar, and salt until well combined.
Add about two thirds of the flour and stir with a spoon or dough hook until a thick, sticky batter forms.
Add the very soft 1/3 cup butter a spoonful at a time, mixing it into the dough until it disappears and the dough looks glossy.
Sprinkle in the remaining flour and knead by hand or with a stand mixer on medium speed for 8 to 10 minutes, until the dough is smooth, elastic, and slightly tacky but not wet.
Shape the dough into a ball, place it in a lightly greased bowl, cover with plastic wrap or a clean towel, and let rise in a warm place until doubled in size, about 60 to 90 minutes.
While the dough rises, make the strawberry filling by combining the chopped strawberries, 1/4 cup granulated sugar, and lemon juice in a medium saucepan.
Cook the strawberry mixture over medium heat, stirring often, until the berries break down and the mixture thickens and bubbles, about 10 to 15 minutes.
Continue to cook until the filling is thicker than jam and you can drag a spoon through the center and the line briefly holds, then remove from heat.
Transfer the strawberry filling to a bowl, cool to room temperature, then chill until cold; reserve 2 tablespoons of the smoothest part of the puree for the glaze.
When the dough has doubled, gently punch it down and turn it out onto a lightly floured surface.
Roll the dough into a rectangle about 12 by 18 inches, with the long side facing you.
Spread the cooled strawberry filling evenly over the dough, leaving a 1/2 inch border along one long edge.
Starting from the opposite long edge, roll the dough into a snug log, pulling gently as you roll to keep the spiral tight, then pinch the seam to seal.
Use a sharp knife or unflavored dental floss to cut the log into 12 equal rolls.
Arrange the rolls cut side up in a greased 9 by 13 inch baking dish, spacing them evenly.
Cover the pan loosely and let the rolls rise in a warm place until puffy and nearly touching, about 30 to 45 minutes.
While the rolls rise, preheat the oven to 350°F and place a rack in the center of the oven.
Bake the risen rolls for 22 to 25 minutes, until lightly golden on top and the centers spring back when gently pressed.
While the rolls bake, make the glaze by beating the cream cheese and 3 tablespoons softened butter in a bowl until smooth and creamy.
Add the powdered sugar and vanilla bean paste and beat until smooth, then mix in the reserved 2 tablespoons cold strawberry puree until fully combined.
Let the baked rolls cool in the pan for 10 to 15 minutes, then spread the strawberry cream cheese glaze over the warm rolls so it melts into the swirls.
Serve warm or at room temperature.