For the queso sauce:
Melt the butter in a small saucepan over medium heat. Whisk in the flour and cook for 1 minute. Add the milk, increase the heat to high and cook, whisking constantly, until slightly thickened, about 5 minutes. Remove from the heat and whisk in the cheese until melted; add the Parmesan and season with salt and pepper, to taste. Keep warm.
For the relish:
Combine all ingredients in a bowl and season with salt and pepper, to taste.
For the pickled red onions:
Bring vinegar, water, sugar and salt to a boil in a small saucepan over medium heat. Remove from the heat and let cool for 10 minutes. Put the onions in a medium bowl, pour the vinegar over, cover and refrigerate for at least 4 hours and up to 48 hours before serving.
For the burgers:
Heat a griddle or large saute pan over high heat. Add the oil and let it heat until it begins to shimmer.
Shape the ground beef with your hands into 4 round patties about 1 1/2 inches thick and season each burger on both sides with salt and pepper, to taste. Cook the burgers until golden brown on both sides and cooked to medium, about 8 minutes.
Put the burgers on the buns and top each with a few tablespoons of the queso sauce, relish, onions and chips.