Combine warm water, yeast and sugar in a large mixing bowl. Beat in milk, melted butter, salt, eggs, and 1 cup of the flour. Beat till smooth. Add enough flour to make a soft dough. Knead till smooth.
Let dough rest for 5 minutes while you prepare the filling.
Beat cream cheese till soft and fluffy. Add ¾ cup powdered sugar and beat till smooth. SET ASIDE ABOUT ⅓ CUP OF FILLING. In another small bowl, combine the jam and diced strawberries.
Roll out dough on a greased counter top to a rectangle about 10x15". Spread on the cream cheese mixture, then the berries. Roll up and pinch the edge to seal.
Cut log into 12 slices and place in a well greased 9x13" pan. Cover tightly with plastic wrap and place in the fridge.
Add the ½ cup of powdered sugar, milk, and vanilla to the reserved cream cheese mixture. Beat till smooth, adding enough milk to get a thin glaze. Then place icing in a ziplock bag in the fridge.
Let rolls chill in the refrigerator for 8-10 hours or overnight.
Remove rolls and bag of icing from the fridge. Place the rolls in a warm spot and let them rise till doubled (about an hour.)
Bake at 350° for 30-35 minutes, or till done in the middle. You may need to cover them with foil to prevent over-browning.
Cut a corner in the ziplock bag and glaze rolls while still warm.