Wash and dry all produce. Adjust oven rack to upper position and preheat oven to 450 degrees. Mince or grate garlic until you have 1/2 tsp (you’ll have a clove left over). Halve, peel, and thinly slice onion. Cut broccoli into florets.

In a small bowl, mix together 3 TBSP mayonnaise (we sent more) and as much of the garlic as you like (start with a pinch and go up from there). Season with salt and pepper.

Heat a large drizzle of oil in a large pan over medium heat. Add onion and cook, tossing occasionally, until browned, 8-10 minutes. Stir in 2 TBSP vinegar (we sent more) and 2 tsp sugar. Continue cooking until syrupy, 1-2 minutes more. Season with salt and pepper. Remove from pan and set aside in another small bowl. Wash out pan.

While onion cooks, toss together broccoli and 1 TBSP aioli in a medium bowl until florets are evenly coated. Add panko, oregano, and a large pinch of salt and pepper and toss so that crumbs stick to broccoli. TIP: It’s OK if not all the breadcrumbs stick. Spread broccoli out on a lightly oiled baking sheet. Bake in oven, tossing halfway through, until panko is golden brown, about 15 minutes.

Heat a large drizzle of oil in pan used for onion over medium-high heat. Shape beef into four patties (make them slightly wider than the buns). Season with salt and pepper. Add to pan and cook until patties almost reach desired doneness, 3-5 minutes per side. Sprinkle patties evenly with cheese. Cover pan and continue cooking until cheese melts, about 1 minute.

While patties cook, split buns in half and place on another baking sheet. Toast in oven until golden, about 3 minutes. Spread ketchup and remaining aioli onto buns. Fill each with a beef patty and as much onion jam as you like. Serve with broccoli on the side.