Preheat oven to 425F.
For the ketchup, heat a saucepot with 2 tablespoons EVOO over medium heat. Add onions, garlic, bay and season with salt and pepper. Cook to soften, 8-10 minutes; add smoked and hot paprika or chili powder, and stir a minute more. Add flour, stir to combine, whisk in vinegar then stock. Add tomato sauce, simmer 10 minutes. Transfer to food processor and pure until smooth. Rinse processor bowl and return to base.
For the fries, cut potatoes into 10 wedges each, dress them lightly with EVOO and season with salt and pepper. Roast on a baking sheet 40-45 minutes, turning once, until deeply golden and crispy.
For the burgers, heat EVOO in a small skillet over medium heat. Add butter and melt it into oil; add onions and cook until tender, stirring frequently, 15 minutes. Season with salt and pepper, and add chicken stock. Keep warm over low heat.
Place chorizo in the food processor and pulse to finely chop; transfer to a bowl. Add beef and season with a little salt and aggressively with black pepper. Combine meats and form 4 patties that are thinner at center and thicker at edges.
Heat a cast-iron skillet or griddle over medium-high heat, drizzle with EVOO, about 1 tablespoon, and cook patties about 10 minutes turning once.
Remove fries from oven and switch on broiler.
Char split rolls under broiler to deeply golden. Rub hot toasted roll surfaces with cut garlic then rub tomato flesh into rolls. Season with salt, pepper, thyme and parsley.
Serve burgers on rolls with onions on top, fries and ketchup alongside.