Grate the butter into the dry ingredients, or cut it in with a fork. Mix until evenly combined.
Add milk and sourdough starter then mix with a spoon or your hands until a dough forms.
Cover the dough and refrigerate for 6 hours to ferment.
After 6 hours, roll the dough into a 12" by 6" rectangle on a floured surface.
To laminate the dough, fold the 12" sides inward to meet in the middle, then do the same with the 6" sides to create an envelope shape. Fold it in half again and roll it out into a 12" by 6" rectangle. Repeat this process two more times for a total of 3 rounds.
Place the folded dough in the covered bowl and return it to the fridge for 2 more hours.
Preheat the oven to 450 degrees Fahrenheit.
Roll the dough out on a floured surface into a 12" by 12" square about 1/4" thick.
Cut the rolled-out dough into 9 squares measuring 4" by 4".
Add 1 1/2 tablespoons of apple filling and 1 1/2 tablespoons of cream cheese filling to the center of each square.
Take the corners on the right side of each square and fold them together and twist. Repeat this with the left two corners to create an open-pocket Danish.
Repeat step 12 for each of the 9 squares.
Bake in the preheated oven for 12-14 minutes at 450 degrees, or until the dough is lightly crisp, flaky, and the bottoms are brown. Make sure to place your oven rack in the center to prevent burning on the bottoms of each Danish.
Apple Filling
Rinse, pat dry, peel, and core the apples.
Cut the apples into chunks.
In a pot, combine apple chunks, cane sugar, cinnamon, and 1/2 cup of water.
Cook on the stovetop over medium heat, uncovered, for 15 minutes, stirring occasionally to prevent burning.
Once the apples are soft and fully cooked, stir in the cornstarch slurry (1 Tbsp cornstarch + 1 Tbsp water) and cook for an additional 1 minute.
Let the filling cool and store it in an airtight container in the fridge until needed.
Cream Cheese Filling
In a bowl, combine softened cream cheese, milk, and honey. Mix with a hand mixer at medium speed.
Cover and store the cream cheese filling in the fridge until needed.