Place the beef in a bowl and season with the salt and pepper. Mix with your hands to incorporate, forming 4 to 6 patties, whichever size you wish and whatever size fits your bun. Heat a large skillet (or the grill, your preference) over medium heat and add the olive oil and butter. Cook the patties until browned on both sides and your desired doneness is reached. I do about 3 to 4 minutes per side for medium-well, but it will depend on how think your burgers are.
In a bowl, mix together the tomatoes, onion, cilantro and lime juiced, stirring well. Season with a pinch of salt and pepper.
To assemble your burgers, cover the bottom bun with avocado slices. Place the burger on top and drizzle on lots of queso cheese. Top with the pico and the jalapeños. Serve immediately.
Heat a saucepan over medium heat and add the olive oil and butter. Stir in the onions and garlic. Season with the salt and pepper. Cook until the onion softens, about 5 to 6 minutes. Slowly stream in 3/4 cup of the half and half, whisking the entire time. Toss the grated cheese with 1 tablespoon of the cornstarch. In a bowl, whisk together the remaining 1/4 cup half and half and 1 tablespoon cornstarch until no lumps remain to create a slurry. Stir the slurry into the saucepan and cook for a minute until the milk thickens. Reduce the heat to low.
Stir in the grated cheese, one small handful at a time, until melted. Transfer the mixture to a crock, larger bowl or a small crockpot heating on low. Spoon on the burgers and serve the extra with chips!