In a glass measuring cup, combine the water, yeast, and sugar; set aside to foam for about 5-10 minutes.
In the bowl of an electric mixer fitted with the whisk attachment, beat the pudding mix and milk together until combined and thickened. Add the butter, Greek yogurt, egg, and salt, beating until fully combined.
Switch to the dough hook, and add the yeast mixture and 3 cups of the flour. Mix the dough on low and allow it to knead for about 6-7 minutes. If dough is still too sticky, add more flour, 1 Tablespoon at a time, until dough just cleans the bowl, but is very soft. It should still stick to the bottom of the bowl.
Coat a large bowl with nonstick cooking spray and place dough in bowl. Seal tightly with plastic wrap. Place bowl in a warm place and allow dough to rise until doubled in size, about 2 hours. In the meantime, spray a 9-by-13-inch baking dish with nonstick cooking spray; set aside.
After dough has doubled in size, punch it down. Turn dough out onto a floured Silpat or floured countertop. With a rolling pin, roll it out to an 18-by-12-inch rectangle. Sprinkle extra flour onto the dough if it begins to stick to the rolling pin.
Make the cinnamon brown sugar filling by combining the butter, brown sugar and cinnamon together in a bowl. Spread the filling evenly over the entire surface of the dough. Pour the blackberry filling on top of the brown sugar filling and spread evenly. Roll dough tightly into a log, ending with the seam side down. Pinch the ends to seal.
Using plain, unwaxed dental floss (or a bench scraper), slice the rolls into 12 evenly-sized rolls. Arrange the rolls cut side down in the prepared baking dish. Cover tightly with plastic wrap.