Combine the water, sugar and yeast iIn the bowl of a stand mixer fitted with the paddle attachment, combine the yeast, sugar, and warm water. Allow to stand until bubbly, about 5 minutes. Add 2 cups flour, the salt, and mix on low speed until well combined, about 2 minutes. Switch to the dough hook attachment. Add the remaining flour gradually and mix on low speed until fully incorporated, about 4 minutes. Turn dough out onto a lightly floured surface. Knead until the dough is heavy, smooth, and not sticky, about 5 minutes. Lightly oil a bowl, place the dough in the bowl, cover loosely and set aside to rise until doubled, about 1 hour.
In a medium bowl, combine the strawberries and blueberries. Add the softened cream cheese and strawberry jam and mix until well combined. Set aside.
Heat oven to 375 degrees F. Divide dough into 12 pieces. Cook Tip: To portion the dough evenly roll the dough into a log shape. Cut the log shaped dough in half, then cut the two halves in half leaving you with 4 equal pieces. Cut those pieces into thirds which will give you 12 even portions. Take one piece of dough and flatten it into a 4 inch circle. Place 2 1/2 tablespoons cream cheese mixture (or filling variation - see below) in the center. Fold up the ends of the dough and pinch closed like an empanada or potsticker. Tuck the ends into the center and twist to form a nice ball. Be careful not to make the bottom of the roll a lot thicker than the top to keep the dough even around the filling. Place the roll, seam side down, on a baking sheet lined with parchment paper. Repeat with the remaining rolls. Beat the egg in a small bowl with 1 tablespoon water. Brush the rolls with the egg wash and sprinkle with turbinado sugar. Bake in the preheated 375 degrees F oven until golden brown, 20 to 22 minutes. Allow to cool prior to serving
Add one 6-ounce package blackberries, sugar, instant tapioca and water to a small saucepan. Bring to a boil over medium heat and cook until saucy, about 10 minutes. Remove the syrup off the heat. Put the remaining blackberries into a large bowl and pour the syrup. Toss to combine and set aside to cool prior to filling bagel dough.