Make the dough: Warm the milk in the microwave for 30 seconds to 110°F. Add the yeast and let sit for 5 minutes to proof.
In a large bowl or stand mixer, combine the warmed milk/yeast mixture with the egg, sugar, butter, flour and salt. Mix until a shaggy dough forms.
Knead the dough for 5 minutes until smooth and elastic. Place in a greased bowl, cover and let rise for 1 hour or until doubled in size.
Punch down the dough and roll it out on a lightly floured surface into a 12x18 inch rectangle.
Spread the strawberry jam evenly over the dough, leaving a 1/2 inch border. Sprinkle the diced strawberries evenly over the jam.
Starting with a long side, tightly roll up the dough into a log. Slice into 8-10 equal pieces and place cut-side up in a greased 9x13 baking pan.
Cover and let rise for 30 minutes while you preheat the oven to 350°F.
Bake for 25-30 minutes until lightly golden brown.
Make the icing by beating together the cream cheese, butter, vanilla and powdered sugar until smooth.
Drizzle the warm rolls generously with the cream cheese icing. Best served warm!