Place cheddar cheese sticks onto small tray or plate (not touching), cover with plastic wrap and put into freezer to chill. (This helps prevent leakage while baking.)
In a large measuring cup, combine warm milk and sugar and mix well. Add yeast and let sit for 5 minutes, until foamy.Meanwhile, in a large mixing bowl combine salt, oil, and 4 cups flour. Once the milk and yeast have proofed for 5 minutes, pour into flour mixture.
Mix in a stand mixer with a dough hook or by hand for 5 to 8 minutes, until dough becomes smooth. Dough should not be very sticky and should pull away from the sides of the bowl easily. (If it is very sticky, add flour 1 or 2 Tablespoons at a time (up to ½ cup) and mix until smooth and pulls away from the sides of the bowl easily.)
Place dough in oiled bowl and cover with plastic wrap for 1 hour or until dough has doubled in size.While dough is rising, prepare 2-3 baking sheets with parchment paper, prepare the baking soda bath by combining water, baking soda, and salt in a large pot and bring to a slow boil, and preheat the oven to 450 F°.
Use ⅓ cup measuring cup to divide dough into 13-14 equal pieces. I don't divide all at once, I prefer to roll out one pretzel at a time so the dough doesn't dry out too fast.
Roll pieces into 4x8 inch rectangles. Place cheddar cheese stick in the middle towards the top of the rectangle. Fold sides over the ends of the cheese stick. Then roll the cheese stick sleeping bag style into the dough. Roll gently a few times to seal the ends and bottom shut.
Gently lower the pretzels one by one into the baking soda bath and boil for 20-30 seconds. Lift them out with a flat spatula and place on the parchment paper prepared baking sheets.
Brush with beaten egg, sprinkle with everything bagel seasoning (or salt) if preferred, and bake for 8 minutes (no longer than 10) at 450 degrees F° until golden brown. Cheese may leak a little, this is fine.
Best served warm, but can be saved in the refrigerator for 1-2 days and reheated in the oven or microwave.