In a medium saucepan, add the diced apples, 1 tbsp. sugar, 1 tbsp. butter, 1 tbsp. cinnamon, nutmeg and cardamom. Cook over medium heat until reduced and apples are very soft, about 20 minutes. Using a fork or potato masher, finely mash the apples. It should be the consistency of chunky applesauce. Let cool while you prepare the yeast.
In a small saucepan heat the soy milk and 2 tbsp. butter until it reaches 110 degrees F. If you don’t have a thermometer, it should be warm to the touch but not hot.
Transfer the milk to a large mixing bowl or bowl of a stand mixer.
Sprinkle on yeast and let activate for 10 minutes. Then add 1 Tbsp sugar, salt and stir.
Next add in 1/2 cup of the apple mixture and stir.
Add flour 1/2 cup at a time, mixing as you go. The dough will be sticky, I’ve made these twice and usually use a little more than 2 1/2 cups. When it is too thick to stir, transfer to a lightly floured surface and knead until it forms a loose dough ball. Add a little flour at at a time if needed but only use as much flour as it will take.
Wipe out the large mixing bowl and spray or coat with oil. Place the dough ball in the bowl and cover with a towel. Place in a warm area to rise until doubled in size, about 1 hour. (I usually place on top my stove with the oven on warm or on top of a radiator. One of the few perks of living in a very old house.)
Place the remaining butter in a small saucepan and melt.
On a lightly floured surface or pastry board, roll out the dough into a long rectangle ~1/4 inch thick. Brush with the melted vegan butter and top with 1/4 cup sugar, 1/2 tbsp. cinnamon and ~3-4 tablespoons of the remaining apple mixture. The apple mixture should be in a very thin layer. You may not need all the apple mixture as you want the dough to be easy to roll and not too heavy. Sprinkle on 1/4 cup of the finely chopped pecans.
Slowly and tightly roll up the dough and situate seam side down. With a serrated knife, cut the dough into 2 inch sections and place in a greased 8×8 square or round pan.
Brush the tops with remaining butter and sprinkle with remaining pecans.
Loosely place a kitchen towel on top and let rise again, about 20 minutes.
Bake! Preheat the oven to 350 degrees F. Bake rolls for 30 minutes until slightly golden brown. Let cool for a few minutes, then frost.
Place the powdered sugar into a bowl. Slowly add in milk, 1/2 tsp. at a time until the frosting is creamy and thick. These taste best warm!