If using frozen tofu, thaw it overnight and squeeze out as much water as possible using your hands or a clean cloth to achieve a sponge-like texture.
Drain and crumble the 4 oz firm tofu into small pieces by pulsing in a food processor until crumb-like; avoid turning it into a paste.
Heat 1 Tbsp olive oil in a skillet over medium heat. Add the finely chopped 1/2 onion and 2 minced garlic cloves and cook on medium-low until soft and translucent, about 3–5 minutes; do not brown.
Transfer the cooked onion and garlic to a bowl and blend with an immersion blender (or finely mince) to make a paste.
In a large mixing bowl combine the crumbled tofu, 1/2 cup breadcrumbs, 1/4 cup dairy-free milk, 4 Tbsp cornstarch, onion-garlic paste, remaining 2 Tbsp olive oil, 1 tsp vegan Worcestershire sauce, 1/2 tsp smoked paprika, 1/2 tsp cumin, 1/2 tsp mustard powder, 1/2 tsp fennel, 1/2 tsp salt, and 1 Tbsp nutritional yeast. Mix thoroughly with a spatula or your hands until evenly combined.
Form the mixture into 4 patties by pressing and shaping with your hands. Firmly compact each patty so it holds together while cooking.
Heat a non-stick skillet over medium heat and add a little oil if needed. Fry the patties 2–4 minutes per side, until nicely browned and heated through.
Toast or lightly grill the burger buns in a pan until slightly charred and warmed.
Assemble each burger: place a patty on the bottom bun, top with a vegan cheese slice (use one or two patties and cheese slices as desired), then add pickled cucumber slices, a drizzle of ketchup, raw onion slices, tomato slices, and ruffled lettuce.
Spread vegan mayo on the top bun, place it on the assembled burger, and serve immediately.