Combine the yogurt, salt, and sumac in a medium bowl or container. Add the chicken and make sure each piece is totally coated. Cover and refrigerate for at least 1 hour or up to 12.
When you’re ready to cook the chicken, make the cucumber salad. Place the cucumbers on a cutting board and smack each one with the side of a chef’s knife or rolling pin until it splits like someone sat on them. Now roughly chop. Add to a bowl with the onion and salt, and toss to combine.
Heat a large cast-iron skillet over medium heat. Add 1 tablespoon oil, then the chopped onion and a pinch of salt. Sauté, stirring occasionally, until the onions are mostly translucent and beginning to brown, about 5 minutes.
While the onions are cooking, remove the chicken from its marinade and scrape away any yogurt clinging to the meat (this can cause the chicken to burn or stick in the pan). Discard the marinade.
When the onions are done, transfer them to a plate, then, immediately add the remaining tablespoon olive oil to the pan. Now add the chicken (it should sizzle). Cook for 3 to 5 minutes per side until the chicken is browned all over and cooked through (about 165°F internal temp). If the chicken feels stuck when you go to flip it, give it a little more time to form a crust; a spatula versus tongs also helps.
While the chicken is cooking, spread the ¾ cup of yogurt around on a serving plate—this is where you’ll build the cucumber salad.
By now, the cucumber salad should have accumulated a lot of cucumbery juices. Good! Add the olive oil and stir. Taste and adjust the salt if needed. Pour the cucumber-onion salad on top of the swirled yogurt. Sprinkle some sumac on top.
Plate the cooked chicken on top of the onions, and serve both dishes together.