Mayonnaise -- In a small bowl, add the green chilies mayonnaise, garlic and lime juice and mix well. If you want, you can blend this in a small food processor or blender, but I actually like a little texture to mine. Cover and refrigerate until ready to use. This can easily be made ahead of time.
Burgers -- In a medium size bowl, add the salsa, chili powder, cumin, grated onion (and yes grate the onion right over the bowl, the juice is important), pinch of salt and pepper. If you use a mild salsa you can add a bit more chili powder, but I use a medium heat salsa which works just fine for this. Mix everything well. Then add in the turkey and mix lightly, I prefer to use my hands; and make sure not to over work the burgers.
Make 4 nice size burgers and place on a small sheet pan lined with either foil or parchment. Make a small indentation in the center of the burger with your thumb. Not too deep, just a little indent; this helps the burger not to shrink up as it grills. Season with salt and pepper and put in the freezer. We are not freezing them, just firming them up. They cook better after they rest a bit and are chilled. About 15-20 minutes in the freezer. These also can be made a head of time.
Garnish -- Prepare your avocado slices and drizzle with the remaining lime juice (just so they doesn't brown), lettuce leaves and cheese.
Grill -- I prefer an outside grill, but a inside grill pan or even a heavy saute pan will work just fine. Spray the pan or grill with a non stick spray and cook the burgers on the first side until you get a nice sear and golden brown, depending on how thick you make them, about 5 minutes. Flip, and make sure when you flip, the burger releases from the pan or grill easily. If you have to scrape to flip it, it isn't ready to be flipped. The second side again will take about 5 minutes, maybe a bit less. Add 2 slices of cheese per burger and melt. As the cheese melts, add the burger rolls either to a broiler or grill to lightly brown.
Finish -- Top the bottom bun with a thin slather of the mayonnaise, then a lettuce leaf; then the burger with melted cheese, another lettuce leaf, avocado slices and top with the second bun slathered with the mayonnaise.
Serve -- ENJOY! A crisp green salad goes great with this; fries of course or chips and a pickle.