Place chickpeas in a large bowl. Use a potato masher to thoroughly mash, while maintaining somewhat of a course texture. Alternately, pulse them in the food processor for 5-10 seconds.
Combine mashed chickpeas with quinoa, egg, garlic, parsley, 1 Tbsp. of the lemon juice, turmeric, cumin, 1/2 tsp. of the salt, and 1/4 tsp. of the black pepper. Mix to combine. Carefully form mixture into 10 golfball-sized patties. Place on a piece of parchment paper.
Heat oil in a large nonstick skillet over medium. Working in batches, cook falafel 2 to 3 minutes per side, until golden brown and crispy. Transfer to a plate.
Combine yogurt, feta, green onion, remaining 1 Tbsp. lemon, 1/4 tsp. salt, and 1/4 tsp. black pepper in a bowl. Mix until smooth.
Serve falafel with yogurt dip in a pita or over mixed greens.