Add 1 cup of all-purpose flour and 1 cup of bread flour along with the sugar, yeast, and salt to a large mixing bowl and whisk together.
Add the water and butter to a microwave-safe bowl and heat, in 45-second increments until warm. Don’t worry if the butter hasn’t melted all the way but it should be softened.
Then add the butter/water mixture to the mixing bowl along with the egg and begin mixing together with a wooden spoon.
Stir in another cup each of all-purpose flour and bread flour and allow the dough to become super thick and sticky.
Add in the final ½ cup of all-purpose flour and mix again. You should have a lot more difficult this time so start using your hands to really bring the dough together and knead it.
Then add the final ½ cup of bread flour to the sticky dough and continue kneading. Once it is smooth and soft (not sticky), it is ready. Shape and roll into a ball.
Rest the dough uncovered for about 10 minutes.
The dough should have gotten a bit bigger. At this point, add the dough to a lightly floured surface and roll it into a rectangular shape. Roll it out to about a 10" x 16" rectangle.
For the Filling:
Spread the strawberry jam over the dough evenly. Evenly arrange the strawberries over the jam and sprinkle with the cinnamon.
Start tightly rolling the dough into a log, starting from the far side towards you. When you reach the end make sure to pinch the sides together well, so the jam doesn’t escape.
Using a piece of unflavored dental floss, cut the log into 12 even slices by sliding the floss under the dough then bringing the ends of the floss up and crossing it over to cut the rolls. Do this by starting to split it in the center then cutting the two pieces into 4 then cutting the 4 into three leaving 12 equal rolls.
Add the rolls directly to a greased 10" x 14" sized baking pan with the swirl side facing up separated by about half an inch. If you want, you can tuck a few bits of strawberry right on the top of the rolls in the filling so you can see them after they bake.
Cover the rolls with a towel and put them in a warm spot to let the rolls rise for an hour or up to an hour and a half, allowing them to proof until they are doubled in size and fill the entire pan. You can also use a proof setting on your oven for about 30–45 minutes to speed things up.
Preheat the oven to 350°F. Bake for 25–30 minutes or until they are golden, then remove and allow them to cool.
For the Frosting:
While the rolls cool, work on the icing.
In a medium bowl, whisk together the melted butter, confectioners' sugar, 4 tablespoons of milk, and vanilla extract. Start with 4 tablespoons of milk and add more as needed to thin it to your liking. Generously spread over the cooled cinnamon rolls, top each roll with a couple of sliced berries and serve. Best enjoyed warm!