In a small bowl, stir together the warm milk, 1 teaspoon sugar, and the yeast. Let stand until bubbly, 5 to 10 minutes.
In the bowl of a stand mixer, add the remaining 1/3 cup sugar, flour, butter, salt, and egg. Using the paddle attachment, mix on medium-low speed until the mixture resembles coarse crumbs. Switch to the dough hook attachment.
Add the milk mixture. Beat on medium-low speed until a smooth and sticky dough forms, about 8 minutes. (The dough will be tacky but shouldn’t stick to a clean finger when quickly touched. If the dough is sticking to your fingers, add additional flour as needed, mixing in 1 tablespoon at a time.)
Lightly oil a large bowl. Scrape the dough into the oiled bowl, and turn the dough to coat. Cover, and let rise in a warm spot until doubled in size, about 1 hour.
On a lightly floured surface, turn out the risen dough. Press the dough with your hands to shape it into a square and release a little bit of air. Roll the dough into a 12x16-inch rectangle.
Lightly butter a 10-inch pie dish or spray with baking spray.
In a small bowl, stir together the strawberries and jam. Spread the mixture over the dough, leaving a 1/2-inch border on one short side.
Starting at the short side opposite the border, loosely roll up the dough into a log, pinching the seam to seal. Using a serrated knife or unflavored dental floss, cut 1/2-inch off each end of the log and discard. Cut the log into 8 rolls.
Place each roll, cut side down, in the grease pie dish. Loosely cover and let rise in a warm spot until doubled in size, about 45 minutes to 1 hour.
While rising, preheat the oven to 350°F.
Uncover and bake for 25 to 30 minutes or until golden brown. Let cool while making the icing.
In a medium bowl, beat the cream cheese and butter until well combined. Add the powdered sugar, milk, and vanilla. Beat until smooth. Spread over the top of the warm rolls.