In a small bowl, mix together yeast and 2 Tbsp. sugar in 1 1/2 cup warm water. Let stand until foamy and creamy (about 5-10 minutes). Add oil to yeast mixture and set aside.
In a stand mixer bowl, or large bowl combine flour and salt.
Pour yeast mixture into flour mixture and knead on low speed for 10 minutes. (If mixing by hand, knead dough for same amount of time.)
Place dough in a lightly oiled bowl, turning to coat. cover with plastic wrap and let rise in a warm place for 1 hour or until doubled in size.
While dough is rising, cut cheese into 1"-long rectangular pieces (I did this by cutting a 1 Lb. block of cheese into fourths, lengthwise, and then cut each fourth in half again lengthwise to make 8 long sticks. Then I cut them down into smaller pieces (see picture above)). Each pretzel will have the equivalent of 1 "stick" of cheese.
Cover cheese to keep from drying out.
When dough has risen, turn 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 out. 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, see picture above).
Pinch dough tightly up around the cheese and roll out until rope shape again. Make sure there are no holes along the seam-- you don't want the cheese to ooze out.
Twist into a pretzel shape. Cover with plastic wrap if needed to keep from drying out.
Preheat oven to 450 degrees. Place foil over a baking sheet and spray with cooking spray.
In a large saucepan, dissolve 1/3 cup baking soda in 4 cups water and bring to a boil, stirring until soda is dissolved. Remove from heat.
Dip each pretzel into the baking soda-water solution for 10-20 seconds. Using a slotted spatula, remove from water and place pretzels on baking sheet. Pinch together any separated seams if needed.
In a small bowl, beat egg with 1 Tbsp. water until foamy. Brush egg mixture over the top of each pretzel and sprinkle with cheese or sea salt.
Bake in preheated oven until golden brown, about 8-10 minutes.