Oil the sides of a medium-sized mixing bowl with the vegetable oil and set aside. In the bowl of a stand mixer, combine yeast, honey, and warm water. Lightly stir and let sit for 5-10 minutes until frothy.
Once the yeast is activated, add the beer, melted butter, and salt. Mix on low speed with a dough hook until combined.
In a separate bowl or measuring cup, whisk together the all purpose and whole wheat flour. With the dough hook still on low speed, add the flour mixture to the wet ingredients, a half a cup at a time. Increase the speed to medium and knead until the dough is smooth and pulls away from the sides of the bowl. Transfer to the well-oiled bowl, cover with a clean kitchen towel, and let rise for one hour in a warm place.
When the dough has nearly doubled in size, preheat the oven to 425°F and bring the large pot of water to a low boil. Cover a baking sheet with parchment paper and prepare the egg wash.
Break off a piece of dough a little larger than a golf ball, and shape it into a rope about an inch thick and 15 inches long, tapering off at the ends. With a sharp knife, slice a three- to four-inch slit half way through the middle of the rope, careful not to cut all the way through. Place a strip of cheese inside the slit and reform the dough around it. Fold the rope in half (with the cheese-filled part in the center of the curve) and twist the ends together.
Add the baking soda to the boiling water. Drop in the twists 2 or 3 at a time and boil for 30 seconds. Remove the pretzels with a spider or slotted spoon and set on paper towels to drain.
Once they've all had a baking soda bath, arrange the pretzels on the prepared baking sheet and brush the tops of each with the egg wash. Sprinkle with pretzel salt to your liking and bake for 10-15 minutes, until the tops are golden. Some cheese might bubble out of the twists, but that’s okay. Serve the soft pretzel twists immediately, with a side of coarse mustard or thick cheese sauce.