In a small saucepan, combine water, milk, and butter. Warm until butter melts. Let cool to about 110°F (warm but not hot).
In a large bowl, mix 2 1/4 cups flour, sugar, salt, and yeast. Add warm liquid and mix. Beat in egg.
Add remaining flour slowly until dough forms a soft ball. Knead for 5-7 minutes until smooth and elastic.
Place dough in a greased bowl, cover, and let rise for 1 hour in a warm spot until doubled in size.
While rising, mix strawberries, sugar, and cornstarch in a small pot. Cook over medium heat until thickened, about 5 minutes. Let cool completely.
Roll dough out on a floured surface into a 12x16 inch rectangle. Spread cooled strawberry filling evenly across the dough.
Roll up tightly from the long side. Slice into 12 equal rolls and place in a greased 9x13 baking dish.
Cover and let rise for another 30 minutes while oven preheats to 350°F.
Bake for 20-22 minutes or until tops are golden brown.
Mix glaze ingredients until smooth. Drizzle over warm rolls and serve.