Combine the yeast, brown sugar and warm water in a mixer bowl fitted with a dough hook attachment. Let it sit for 5 minutes until foamy.
Add the melted butter and salt to the mixture and combine.
Add the all purpose flour while the dough hook is rotating.
Next add the whole wheat flour and continue to mix until the dough has formed a large mass. This will take a few minutes, so be patient.
Remove the dough from the bowl and let rest in an oiled bowl for an hour, covered with a kitchen towel.
Preheat the oven to 425 degrees F.
Place a large pot with 10 cups water on the stove and bring to a boil. Add the baking soda. The baking soda, will make the water bubble, so add it in batches if you're worried about the pot overflowing.
Remove the dough from the bowl and break off a little bit of dough, roughly half the size of a golf ball, and flatten it into a disc. Add 1 cube of pepper jack cheese and fold the dough over, pinching in the edges so the cheese is hidden from view and you have a ball of pretzel dough. Repeat this process for the remaining dough.
Using a slotted spoon, add about 10 of the pretzel bites to the boiling water at a time. Once they rise to the top of the water, should take about 30 seconds, remove with your slotted spoon to a paper towel to dry. Repeat with the remaining pieces of dough.
Transfer the pretzel bites to parchment lined baking sheets that you’ve sprayed with cooking spray.
In a small bowl, whisk together the egg yolk and a splash of water. Brush the tops of each pretzel bite with the egg wash and sprinkle with sea salt.
Transfer to the oven and bake until golden brown, about 10-16 minutes.
Remove from oven and let cool slightly before serving!