Heat a grill or grill pan over medium-high. Grill the scallions until charred in spots, turning occasionally, about 6 minutes. Let cool; coarsely chop.
In a food processor, pulse the walnuts and oats until a coarse meal forms. Add 1 cup lentils and pulse until a thick but not mushy paste forms; transfer to a large bowl. Add the remaining 1 cup lentils, the farro, mashed sweet potato and chopped scallions; season with salt and pepper and mix well. Form into six 3/4-inch-thick patties; season both sides with salt.
Heat a large skillet over medium. Spread some butter on the cut sides of the buns. Working in batches, toast the buns in the skillet until browned, 1 to 2 minutes. Transfer to a work surface. Wipe out the skillet; add the oil and heat over medium-high. Add the patties; cook until browned, 2 to 3 minutes per side. During the last minute of cooking, top each patty with 2 slices cheese; tent with foil to melt cheese. Transfer the patties to paper towels and let rest 5 minutes. 4. Build the burgers with the bun bottoms, scallion mayo, patties, lettuce, zucchini pickles, roasted tomatoes, balsamic onions and bun tops.