Preheat oven to 425 degrees F. Line a 9x13 inch baking sheet with parchment paper or a silpat mat (or leave ungreased).
In the bowl of a standing mixer fitted with the paddle attachment or a large mixing bowl, combine the all-purpose flour, whole-wheat flour, baking powder, baking soda, and salt.
Add butter pieces, ensuring that they are very cold. Mix on low speed, just until the butter resembles peas (if you are having trouble breaking apart the butter, try cutting in by hand with a fork.) You should still be able to clearly see the pieces of butter.
In a separate bowl, combine the apple butter, buttermilk, and egg. Add all at once to flour mixture, and stir by hand (or beat on low) just until the dough begins to come together. Stir in the cheddar and dill, until roughly combined.
Dump batter onto a lightly floured surface and knead by hand about 10 times.
Add a bit more flour to your work surface. Roll or pat dough into a 1-inch thickness (make sure it's 1 inch!). Cut into 2-inch rounds with a lightly-floured biscuit cutter or cut into 2-inch squares with a sharp, lightly floured knife. Combine scraps and reshape to cut more biscuits. Place on prepared baking sheet.
Bake for 12-14 minutes until lightly golden on top. Serve warm with apple butter.
To store: Place biscuits in an air-tight zip-top bag and keep at room temperature for 1-2 days or in the refrigerator for up to 1 week, reheating in the microwave for 20-30 seconds prior to serving. To freeze, wrap individual biscuits tightly in plastic wrap and place in a heavy duty freezer bag for up to 2 months.
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