In a large mixing bowl, combine 1 cup of all-purpose flour and 2 teaspoons of instant yeast. Gradually add ½ cup of warm milk (about 125°F) and stir until mixed. Cover the bowl with a towel and let it rest in a warm place for 30 minutes.

Add in 1 cup of grated apples, 2 tablespoons of softened butter, 2 tablespoons of granulated sugar, 1 large beaten egg, and a pinch of salt to the yeast mixture. Gradually incorporate 2 cups of additional flour until a soft dough forms.

Transfer the dough to a floured surface and knead for about 5-7 minutes until it becomes elastic and smooth. Form the dough into a ball and place it in a lightly buttered bowl, covering it with a kitchen towel. Allow it to rise in a warm area for 45-60 minutes.

Once the dough has risen, roll it out on a floured surface to a rectangle, about ½ inch thick. Spread 3 tablespoons of softened butter evenly over the surface.

In a separate bowl, combine ½ cup of brown sugar, 1 teaspoon of ground cinnamon, ¼ teaspoon of ground nutmeg, 1 cup of grated apples, and ½ cup of crushed pecans. Sprinkle this mixture over the buttered dough.

Starting from one long edge, tightly roll the dough into a log, sealing the edge by pinching it with your fingers. Cut the rolled dough into 12 equal pieces. Arrange these pieces in a buttered muffin tin with the cut sides facing down.

In a saucepan, combine ½ cup of granulated sugar and 2 tablespoons of water over medium heat. Stir gently until the sugar dissolves and starts to bubble. Cook until it reaches a light amber color.

Remove from heat and carefully add ¼ cup of cold butter and ¼ cup of heavy cream, stirring until fully combined.

Place ½ cup of pecan halves in the bottom of each muffin cup, then pour the prepared caramel evenly over them. Gently place the dough pieces on top, ensuring they are cut-side down in the caramel.

Cover the muffin tin tightly with plastic wrap and refrigerate overnight.

Preheat your oven to 350°F when you're ready to bake. Uncover the muffin tin and let the buns rise in a warm spot until they’ve doubled in size, about 30-45 minutes.

Bake for 18-22 minutes or until they are golden brown and fragrant.

Once baked, remove the muffin tin from the oven and let it cool for a few minutes. Carefully invert the tin onto a serving platter, allowing the caramel to drizzle over the sticky buns.