Place ⅓ cup all-purpose flour, ½ cup milk, and ½ water in a saucepan and whisk to combine. Heat over medium heat, while whisking constantly but slowly, and cook until thickened. Remove from the heat and set aside.
Heat the reaming ½ cup milk until 100 to 110°F. You can microwave for 30 to 60 seconds until it is just hot to the touch. Pour the milk into the bowl of a stand mixer along with the yeast and about 2 teaspoons of sugar. Stir to combine and let sit until foamy.
Add in the remaining flour, bread flour, eggs, melted butter, and salt. And in the milk/flour roux and stir to combine.
Use a dough hook to knead until smooth and soft, but not too sticky.
Place the dough in an oiled mixing bowl and cover with plastic wrap. Allow to rise until doubled in size, 90 to 120 minutes.
When ready, lightly dust your work surface with a little bit of flour and roll the dough out into a large rectangle, about 18 X 14-inches.
Spread a thin layer of 2 tablespoons butter over the surface of the dough. Dollop on the jam and spread all over.
Starting at the long edge of the rectangle, carefully roll up the dough into a long log. Slide a piece of dental floss under the log until about halfway up, pull up on the strings, cross them, and pull to cut the dough in half. Continue to cut the log into 12 even pieces. If the dental floss is tricky, use a serrated knife.
Butter the inside of a 9 X 13-inch baking dish. Place the cut cinnamon rolls, cut-side up, in the bottom – leaving about a half-inch space in-between rolls. Cover with a clean kitchen towel and allow to double in size – about 30 to 45 minutes in a warm room.
Meanwhile, pre-heat the oven to 350°F. While you wait, make the strawberry sauce for the frosting (recipe steps to follow)
Remove the kitchen towel and bake the cinnamon rolls for about 25 to 35 minutes, or until slightly golden around the edges and an internal temperature of 190°F. Let cool as you finish the frosting.