Step 1: Prepare the Apples: Peel, core, and chop Granny Smith apples into small, even pieces. Place in a microwave-safe bowl and microwave on HIGH for 2 to 2 ½ minutes to partially cook. Drain any liquid well and set aside to cool completely.
Step 2: Make the Dough: In a large mixing bowl, whisk together 2 cups of the flour, ⅓ cup granulated sugar (for dough), both packets of RapidRise yeast, and salt. In a separate microwave-safe bowl, combine water and 6 tablespoons of butter (for dough). Heat in 15-second bursts until 120°-130°F (49°-54°C). Pour this warm mixture into the dry ingredients along with the large egg.
Step 3: Beat and Knead: Using an electric mixer, beat the dough mixture on medium speed for 2 minutes. Add another 1 cup of flour and beat on high for 2 more minutes. Slowly stir in enough of the remaining flour until a soft, slightly sticky dough ball forms. Turn out onto a lightly floured surface and knead for 6-8 minutes until smooth and elastic. Cover with a clean kitchen towel and let rest for 10 minutes.
Step 4: Make the Filling: While the dough rests, combine ⅓ cup granulated sugar (for filling), 1 teaspoon all-purpose flour (for filling), and ground cinnamon in a small bowl. Stir well and set aside.
Step 5: Roll and Fill: Uncover the dough and roll it out into a 15x10-inch rectangle. Evenly spread 3 tablespoons of very soft butter (for filling) over the dough, leaving a ½-inch border along the long sides. Scatter the cooled, chopped apples over the butter, then generously sprinkle the cinnamon-sugar mixture on top.
Step 6: Form Rolls and Rise: Carefully and tightly roll the dough into a log from one of the long edges. Pinch the seam closed. Slice the log into 12 equal rolls using a sharp knife or unflavored dental floss. Place them cut-side down in a greased 13x9-inch baking pan. Cover with a towel and let rise in a warm spot for about 1 hour, or until doubled in size.
Step 7: Bake: While rolls are rising, preheat your oven to 350°F (175°C). Once risen, bake for 20-30 minutes, or until golden brown.
Step 8: Make Glaze and Glaze Rolls: As soon as rolls come out of the oven, transfer them to a wire rack to cool for at least 20 minutes. In a medium bowl, whisk together powdered sugar, melted butter (for glaze), milk, and pure maple syrup until smooth. Drizzle the glaze over the warm (but not hot!) rolls and serve immediately.