Make the savory Greek yogurt dip: Combine the yogurt, parsley, mint, cumin, lemon zest, salt, and pepper in a medium bowl. Transfer to an airtight container, cover, and refrigerate for up to 10 days.
Make the sliders: Heat a 10-inch cast-iron skillet over high heat, then add the olive oil, mushrooms, and salt. Saute until the mushrooms are soft and golden, approximately 5 minutes. Remove from heat and let cool.
Thoroughly combine the turkey, Parmesan, miso, and cooled mushrooms using a potato masher, or, better yet, your fingers. Divide the mixture into balls a little bigger than the size of a ping-pong ball. Cover and refrigerate for at least 30 minutes.I usually do this overnight.
Return the skillet to medium heat. Flatten the turkey balls so that they're quite thin, about 2 1/2 inches in diameter. Arrange the patties in the skillet, making sure they aren't touching. Cook for 2 minutes, or until browned, then flip and cook for another 2 minutes. Flip again for 1 minute and one final time for 1 minute, for a total of 6 minutes of cooking time.
Transfer the patties to a platter. Place the buns, cut-side down, in the skillet. Cover and cook for 2 minutes, or until brown. Dress a bun bottom with the Savory Greek Yogurt Dip; top with a patty, then lettuce, a tomato slice, and a bun top.