Start by washing your potato thoroughly and grating it using the large holes of your box grater, leaving the skin on for texture if you like. Slice the green onions thinly, using both the white and green parts to maximize flavor and color in the mix. Pat the grated potato dry with a paper towel to remove excess starch and moisture before mixing.
Place your ground chicken in a large mixing bowl and add the shredded cheddar cheese, prepared potato, and chopped green onions. Pour in the BBQ sauce and sprinkle generously with salt and fresh cracked pepper to ensure every bite is seasoned well. Use your hands to gently fold the ingredients together until just combined, being careful not to overwork the meat.
Divide your chicken mixture into four equal portions to ensure they all cook at the same rate on the grill. Gently pat each portion into a round disc that is approximately three-quarters of an inch thick. Use your thumb to create a small indentation in the center of each patty to prevent them from puffing up while cooking.
Preheat your grill or skillet to medium-high heat, ensuring the surface is well-oiled to prevent sticking. Place the patties on the heat source and let them cook undisturbed for about 4 to 5 minutes on the first side. Flip carefully when they release easily from the grate and have developed a nice golden-brown crust.
Brush a little extra BBQ sauce over the cooked side of the burgers while the second side finishes grilling. Check the internal temperature with your thermometer to ensure it reaches exactly 165°F for safety. Remove the burgers from the heat and let them rest on a plate for a few minutes to redistribute juices.