Mix the beef and the eggs in a bowl by hand. The eggs will act as a binder to hold the meat together on the grill.
Divide the ground beef into eight or nine equal cylinders and coat each one in taco seasoning. Place a piece of plastic wrap slightly larger than a cylinder on a cutting board. Coat the plastic with some taco seasoning. Set the first cylinder on the wrap.
Press the cylinder to flatten it and use your thumb to draw a trench down the center of it. Tear a slice of cheese into quarters. Fill the trench with two of the quarters. Use the plastic to roll the ground beef around the cheese. Repeat with all eight or nine cylinders.
If you haven’t already done so, slice the avocado thinly enough to get four slices per taco dog.
Grill each burger dog for 4-6 minutes per side, lid on, just like you would with a burger or hot dog. During the last minute of grilling, toast the buns for 45-60 seconds.
Place each finished dog in its respective bread product. Add the shredded cheese first so it melts, then finish each taco burger dog with the condiments and chow down.