Heat milk in a small saucepan until warm, then mix in butter until melted. Let it cool to lukewarm.
In a large bowl, combine 2 1/4 cups flour, yeast, sugar, and salt. Mix well.
Stir in egg and 1/4 cup water. Mix, then add the milk mixture and stir until dough forms.
Add another cup of flour and stir. When dough forms, turn it onto a floured surface and knead until smooth.
Place the dough ball back in the bowl, cover with a damp towel, and let rest for 10 minutes.
Roll dough into a 15×10 inch rectangle. Spread strawberry jam, sprinkle with cinnamon, and top with chopped strawberries.
Roll up the dough, pinch the seam to seal, and cut into 12 equal rolls.
Place rolls cut-side up in a greased 11×13 dish, cover, and let rise for 30 minutes. Preheat oven to 375°F.
Bake for 30 minutes or until golden brown. For the icing, mix powdered sugar, cinnamon, and milk, then drizzle over the rolls.