Roughly chop the chard leaves, place in a steamer basket and set over a pot filled with 1 inch of water. Bring the water to a simmer, cover and steam for 1 minute, or until the leaves are wilted. When cool to the touch, squeeze the excess water out of the leaves and chop finely.
In a large bowl, combine the chard with the beans, onion, garlic, mayonnaise, mustard, dill, lemon zest, salt, pepper and cayenne. Use a potato masher to mash until the mixture holds together but still has some visible bean chunks. Add the egg and fold until combined. Then fold in the panko. The mixture should be cohesive and a little wet.
Form the mixture into 6 patties. Place the patties on a baking sheet, cover loosely with foil or plastic wrap, and chill overnight.
Make the pickled chard stems: In a lidded jar, combine the chard stems, onion, lemon juice and salt. Shake to combine and chill until ready to use.
Make the lemon dill yogurt: In a small bowl, combine the yogurt, lemon juice, dill and salt. Chill until ready to use.
Preheat the oven to 400°F and line a baking sheet with parchment paper.
Heat a cast-iron skillet over medium heat. Coat the bottom with oil and cook the burgers for 3 to 4 minutes per side, or until browned, working in batches and reducing the heat to low as needed. Transfer the burgers to the prepared baking sheet and bake for 15 minutes. If desired, warm the buns in the oven during the last few minutes of cook time.
Assemble the burgers on the buns with the yogurt sauce, pickled chard stems and desired fixings.