4 large flour tortillas
2 medium-sized chicken breasts
BBQ sauce (I just used Kroger brand original; you might want to use something more spicy)
1-2 avocados, sliced
about 12 oz shredded monterrey jack cheese (pepper jack would also be good if you like it)
1 large yellow onion
butter (have about a stick on hand)
S&P, garlic powder, cumin
Let's start with the onions since they take the longest. Slice the top end off the onion, then peel off the skin. Cut it in half top-to-bottom (cutting through the root end). Putting cut-side-down on your cutting board, slice the whole onion into thin slices (half-circles). Work up to the root and discard the root end. The reason I'm giving these onion-slicing details is that if you work with it cut-side-down and also cut the root end off last, it minimizes the teary eyes. In fact when cutting up an onion this way I don't shed a single tear!
Heat up a medium or large skillet over medium heat; melt about 2T butter and add all the onion slices, breaking them up with a spatula. Cover them and come back to stir them occasionally, cooking until they are all translucent and turning golden brown.
Pre-brown your "top" tortillas. Heat a griddle or large skillet on medium heat with about 1T butter. Spread the butter around evenly and put a tortilla (or two if it'll fit); cook until it starts to crisp up; golden brown.
Chicken: whenever I'm going to cook chicken on the grill I always pound out the breasts first. This makes them more tender and also evens out the thickness so they'll cook more evenly, and more quickly. After pounding the chicken (between two pieces of plastic wrap, with meat mallet), season both sides with salt, pepper, garlic powder, and cumin. You can get creative and use any other spices you'd like. Cook the chicken on the BBQ and brush 2-3 T BBQ sauce on each side as it cooks. When it's cooked through, let it cool off a few minutes and then slice into thin strips.
Assuming the onions are cooked by now, it's time to assemble!
Put some more butter on the skillet (1-2T), and your two fresh (uncooked) tortillas will be the bottom of your quesadillas. Start by sprinkling a big handful of jack cheese on each. Then, several pieces of chicken. Then, sliced avocados. Then, the onions. At this point I brushed on a few more tablespoons of BBQ sauce and sprinkled on a little more cheese to help the "lid" stick. Then, add the pre-browned tortillas; browned side up.
I didn't intend to flip them over as they cooked, but I did so that they would be hot all the way through. I just cooked the first side until the bottom tortilla had browned, then carefully (and quickly) flipped it over with a big spatula and let the other side cook for just a minute.