Heat the milk in a small saucepan until warm. Mix in the softened butter until fully melted. Let the mixture cool to lukewarm.
In a large bowl, combine 2 1/4 cups of flour, instant yeast, sugar, and salt. Mix well.
Stir in the egg and 1/4 cup of water. Mix thoroughly. Then, stir in the milk mixture until a dough forms.
Add the remaining 1 cup of flour and mix until the dough comes together. Turn the dough onto a lightly floured surface and knead until smooth and elastic.
Place the dough ball back into the bowl, cover with a damp towel, and let it rest for 10 minutes.
Roll out the dough into a 15×10 inch rectangle on a floured surface. Spread the strawberry jam evenly over the dough. Sprinkle with cinnamon, then evenly distribute the sliced strawberries on top.
Carefully roll up the dough from one long edge to the other, pinching the seam to seal. Cut the roll into 12 equal-sized slices.
Place the rolls, cut side up, in a greased 11×13 inch baking dish. Cover and let the rolls rise in a warm place until doubled in size, about 30 minutes. Meanwhile, preheat the oven to 375°F (190°C).
Bake the rolls in the preheated oven for 30 minutes, or until golden brown on top.
Icing:
In a small bowl, mix the powdered sugar and cinnamon with 1 tablespoon of milk. Stir until a smooth glaze forms. If needed, add more milk, a little at a time, to reach your desired consistency.
Drizzle the glaze over the warm rolls before serving.