Slice the red onion 3 mm thick and pack it into a 500 ml / 2-cup jar or bowl. Stir the vinegar, water, sugar and 3/4 tsp salt until the sugar dissolves, pour it over the onion, and press the slices under the liquid. Leave it on the counter for at least 20 minutes while you cook.
Stir the yogurt, lemon juice, grated garlic, Dijon, dill and 1/4 tsp salt together, adding water 1 tbsp at a time until the sauce drizzles off a spoon. Refrigerate it. Dice the tomatoes, chop the romaine and slice the cucumber, then set them aside.
Mix the turkey, panko, egg, Worcestershire, smoked paprika, garlic powder, 3/4 tsp salt and pepper with your hands for about 30 seconds, only until combined. Divide into four 150 g portions, shape each into a 10 cm / 4-inch patty about 2 cm thick, and press a shallow dimple into the center with your thumb.
Rinse the rice in a fine-mesh sieve until the water runs mostly clear. Combine it with the water, cumin and 1/4 tsp salt in a 3-quart / 2.8 L saucepan and bring to a boil. Cover, reduce to low and cook for 15 minutes. Scatter the beans over the rice without stirring, replace the lid, and leave it off the heat for 5 minutes.
While the rice simmers, heat the olive oil in a 30 cm / 12-inch cast-iron skillet over medium heat until it shimmers, about 2 minutes. Add the four patties and cook 5 to 6 minutes per side, until deeply browned and the center reads 165 F / 74 C on an instant-read thermometer.
Move the patties to a plate and top each with 15 g shredded cheddar. Rest them for 3 minutes so the cheese melts and the juices settle.
Fluff the rice and beans with a fork and divide them among four bowls. Add the romaine, tomato and cucumber, then top each bowl with a patty, a spoonful of yogurt sauce and a handful of drained pickled onions.