For the queso sauce:
1) 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.
2) 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:
1) Combine all ingredients in a bowl and season with salt and pepper, to taste.
For the pickled red onions:
1) Bring vinegar, water, sugar and salt to the boil in a small saucepan over medium heat. Remove from the heat and let cool for 10 minutes.
2) 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:
1) Heat a griddle or large saute pan over high heat. Add the oil and let it heat until it begins to shimmer.
2) Shape the beef mince with your hands into 4 round patties about 4cm 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.
3) Put the burgers on the buns and top each with a few tablespoons of the queso sauce, relish, onions and crisps.