In a large non-reactive bowl, whisk together yogurt, olive oil, lemon juice, vinegar, minced garlic, and all spices to form a marinade paste.
Add sliced chicken thighs to the bowl and toss until thoroughly coated; cover and refrigerate for a minimum of 4 hours, ideally 12-24 hours.
Prepare the Toum: Place 1 cup garlic cloves and 1 tsp salt in a food processor. Pulse until finely minced, scraping down the sides repeatedly.
With the processor running, very slowly drizzle in 1/2 cup of neutral oil in a thin, steady stream until a paste forms.
Continue the emulsion by alternating between 1 tablespoon of lemon juice and 1/2 cup of oil, adding slowly until all ingredients are incorporated and the sauce is fluffy and white.
Preheat a heavy cast-iron skillet over medium-high heat or an oven to 425°F (220°C).
Add chicken to the skillet in a single layer (cook in batches if necessary) and sear for 8-10 minutes per side until deeply charred and internal temperature reaches 165°F (74°C).
Rest the chicken for 5 minutes before slicing into thin shavings.
Serve immediately on warm pita bread with a generous spread of garlic toum, pickled turnips, and fresh vegetables.