In a large skillet over medium-high heat, brown the ground beef, 5 to 8 minutes. Drain the excess fat. Add the tomato paste, chili powder, cumin, salt, cayenne and 1 cup water and stir to combine. Bring to a boil, lower to a simmer and simmer until thickened, 8 to 10 minutes. Keep warm.
Heat a separate skillet over medium heat. Smear butter on the bottom and place a tortilla in the skillet. Top the tortilla with some of the cheese and a second tortilla and cook until the bottom side is brown and the cheese begins to melt, about 2 minutes. Flip and cook the other side another 1 to 2 minutes. Transfer to a plate and keep warm. Repeat with the remaining tortillas and cheese, adding additional butter as needed.
Top each quesadilla with a generous helping of the beef mixture, then top with some lettuce and Pico de Gallo.
Chop the tomatoes, jalapenos and onions into very small dice (leave the seeds in your jalapenos for a hotter pico, or remove for a milder pico). Next, chop up a nice-size bunch of cilantro. Just remove and discard the long leafless stems before chopping. No need to remove the leaves from the stems completely. Place all of these ingredients together in a bowl and give it a good stir. Squeeze the juice of the lime half into the bowl. Add salt to taste and stir again.