Butterfly and pound chicken breasts to 1/4-inch thickness. Combine chicken and yogurt in a medium bowl, season with salt and pepper, and toss with tongs until chicken is well-coated. Let stand at room temperature 15 minutes (or cover and refrigerate up to 2 hours). Coat chicken with panko, patting to adhere.
Heat 1/2 inch vegetable oil in a large straight-sided skillet (preferably cast iron) over medium-high until a panko crumb sizzles when tossed in. Add half of chicken; cook, rotating with tongs for even browning, until golden brown and cooked through, 3 to 4 minutes a side. Transfer to a wire-rack-lined baking sheet; season with salt. (Remove and discard any bits of coating with a slotted spoon before cooking second batch.) Repeat with remaining chicken.
Place greens in a bowl; drizzle with olive oil and lemon juice and toss to coat. Season with salt and pepper and serve immediately with cutlets and lemon wedges.