Butter the bottom and sides of a 13 x 9-inch baking dish with 1 tablespoon of butter. Line the bottom of the pan with parchment paper. If you’re using a USA Pan baking pan or a similar pan with nonstick coating, you won’t need to line it with parchment.
Cut the remaining butter into 8 pieces. Heat the milk in a small saucepan to 180°F (82℃). Remove the saucepan from the heat, add the butter and gently stir until the butter is almost completely melted and there are just 1 or 2 small pieces left. Pour the mixture into a 1 cup measuring cup and cool it to 120–130°F (49-54°C).
Crack the eggs into a small bowl and whisk them together.
Place 2-1/2 cups of the flour, the salt, yeast, and sugar in the bowl of a stand mixer. When you add the yeast and salt to the bowl, make sure to add them on opposite sides of the bowl. You don’t want the yeast and salt to come in direct contact. Whisk the ingredients together.
With the paddle attachment and the mixer on low speed (speed 2 on a KitchenAid mixer) slowly add the warm milk to the dry ingredients. When all the milk is added slowly pour the eggs into the bowl as the mixer is running.
When the eggs are fully incorporated and there are no bits of flour left, add the remaining flour in 5 batches. You can use a large spoon or a measuring cup for this step. As soon as all the flour is mixed in, switch to the dough hook. Make sure to remove as much dough as possible from the paddle attachment.
With the dough hook knead the dough on low speed (speed 2 on a KitchenAid) until the dough is smooth, elastic and clears the sides of the bowl, 13-15 minutes. The dough should not stick to your finger when you poke it with your finger, and it should pass the windowpane test.
For the windowpane test, take a small piece of dough, about the size of a golf ball and gently stretch it into a square. Keep on stretching it until it becomes a thin enough for light to pass through it without tearing. If the dough tears, knead for 2 more minutes and perform the windowpane test again.
Transfer the dough to a lightly floured counter and knead by hand to form a smooth, round ball, about 30 seconds. Lightly spray a large bowl or an 8-cup glass measuring cup with non-stick spray or brush it with vegetable oil. Place the dough seam side down in the oiled container and cover tightly with plastic wrap.
Place the dough in a warm spot let it rise until doubled in volume, about 60-90 minutes. If you kitchen is cold see my note below for creating your own proofing box.
NOTE: I recommend using the 8-cup measuring cup because the markings on the side are a good indicator of when the dough has doubled.
When the dough has risen gently turn the dough onto a floured surface. Use a rolling pin to roll the dough into a 12 x 19-inch rectangle. As you roll out the dough roll it out from the center, so the dough is even thickness throughout. If the dough keeps springing back on you and is difficult to roll out, let it rest for 10 minutes and then try rolling it out again. Use your hands to gently stretch the corners of the dough so they are as square as possible.
Use an offset spatula spread the cinnamon-sugar filling across the dough and leave an 1/2-inch border. Spread the cooled cooked apples over the cinnamon filling. With the long side of the dough square facing you, roll the dough into a tight cylinder. Roll the dough until the seam is face down. Trim the ends so the cylinder is 18 inches long.
Use a ruler to mark the dough and with a sharp knife cut 12 rolls 1 1/2-inches wide each.
Place the rolls in the buttered pan evenly spaced from each other and the sides of the pan. Cover the rolls with plastic wrap and let rise until doubled in size, about 1 hour. If your kitchen is still cool you can recreate the proofing box in your oven.
When the rolls look doubled in size check them by gently and lightly poking a puffy corner with the knuckle of your index finger. If the dough springs back it is ready.
Preheat the oven to 350°F (180℃). Right before the rolls are placed in the oven, heat 1/2 cup of heavy cream in the microwave until it is about 90°F (32℃). With a pastry brush, brush the top of the rolls with the cream. Slowly pour the remaining cream over the rolls. The cream mixture will settle on the bottom of the pan.
Bake the cinnamon rolls for 30-35 minutes until the cinnamon rolls have risen and they are dark golden brown. Use an instant-read thermometer to check the internal temperature of one of the rolls in the center. It should register 190℉ (88℃).
Remove the cinnamon rolls from the oven and place the pan on a wire rack to cool.
While the rolls are cooling make the cinnamon roll glaze (see the recipe below). Let the rolls cool for 15 minutes, then spread the glaze over the rolls while they are still warm.