Step 1: In a small bowl, mix together yeast, sugar, and warm water. Let stand until foamy and creamy, about 10 minutes. Add oil to the yeast mixture. Set aside.
Step 2: In a stand mixer bowl or large bowl, combine flour and salt.
Step 3: Pour yeast mixture into flour mixture and knead on low speed for 10 minutes. (If mixing it by hand, knead the dough for the same amount of time.)
Step 4: Lightly oil a large bowl. Place dough inside, turning to coat.
Step 5: Cover with plastic wrap and let rise in a warm place for 1 hour, or until doubled in size.
Step 6: While the dough rises, cut the cheese into 1-inch-long rectangular pieces. I did this by cutting the 1-pound block of cheese into fourths lengthwise, and then I cut each fourth in half lengthwise again to make 8 long sticks. (Each pretzel will use 1 of these long sticks of cheese.) Cut each stick into 1-inch-long pieces. Cover the cheese to keep it from drying out. Set aside.
Step 7: When the dough has risen, heat the oven to 450 degrees F. Cover a baking sheet with aluminum foil and spray foil with cooking spray. Set aside.
Step 8: Turn dough out onto a lightly floured surface and divide into 8 equal pieces. Roll each piece into a rope and use a rolling pin to flatten it out.
Step 9: Place cheese pieces along the middle of the dough, leaving a few inches of dough on each end without cheese. This will make it easier to twist into shape.
Step 10: Pinch the dough tightly up around the cheese and roll it out until it is in a rope shape again. Make sure there are no holes along the seam; you don't want the cheese to ooze out.
Step 11: Twist into a pretzel shape. Cover with plastic wrap if needed to keep it from drying out.
Step 12: In a large saucepan, bring baking soda and 4 cups of water to a boil, stirring until the baking soda is dissolved. Remove from heat.
Step 13: Completely dip each pretzel in the baking soda solution for 10-20 seconds each. Using a slotted spatula, remove pretzels from the water and place them on a baking sheet. Pinch together any separated seams, if needed.
Step 14: Preheat oven to 450 degrees. Place aluminum foil over a baking sheet and spray with cooking spray.
Step 15: In a small bowl, beat the egg with the remaining 1 tablespoon water until foamy. Brush egg mixture over the top of each pretzel and sprinkle with cheese or sea salt.
Step 16: Bake for 8-10 minutes, until golden brown.