For the pickled onions: In a pot, combine the vinegar and brown sugar, and bring to a simmer. Whisk to dissolve the sugar. Add the onions and bring the pot back to a simmer and cook until the onions are tender.
Remove from the heat and store the onions in their pickling liquid in a non-reactive container for at least 1 hour, but preferably overnight.
For the burgers: In a bowl mash 1 1/2 pounds of the black beans. Add the remaining black beans to the bowl with the corn, rice, chili powder, cumin and coriander, and mix everything together by hand. Season with salt and pepper. Form the burger mixture into 6 patties and set aside.
For the cream: In a bowl whisk together the sour cream, honey, lime juice and cilantro. Season with salt and pepper, and adjust the flavor with more honey if the dressing is too tart.
For Plating: Heat a cast-iron griddle over high heat until it begins to smoke. Butter both sides of the buns and toast them on the griddle until golden brown.
Heat the grapeseed oil on the griddle. Place the burgers on the griddle and sear the first side for about 3 minutes. Flip and cook the second side until seared and the center is hot, about more 3 minutes.
In a bowl combine the arugula with 2 tablespoons of the prepared sour cream, a squeeze of lime juice and some salt.
Spread the prepared sour cream on each side of the buns. Place the burgers on the bun and top each with 2 ounces of the pickled red onions and 1/2 cup of the arugula salad.
For extra flavor, make the pickled onions the day before you want to serve your burgers.