To make the pesto, combine cilantro, pepitas, lemon juice, and cumin in the bowl of a food processor and pulse to combine. Drizzle in olive oil and honey (if using) and add salt and pepper to taste. Pesto will keep well covered in the fridge for 3 days.
To make burgers, heat a grill or grill pan to medium heat.
Prepare mushrooms by removing stems and cleaning caps with a damp cloth or paper towel. Place mushrooms in a medium bowl and drizzle with olive oil, balsamic vinegar, tamari, and pepper. Toss until well coated. (I use my hands to rub it all over until they are fully coated on both sides.)
Place mushrooms, gill side down, on grill or grill pan. Cook 5 - 7 minutes per side, or until mushrooms are tender.
Place mushrooms on toasted buns and top each with quick-pickled onions, greens, and a generous slather of pepita pesto.