Preheat grill to medium. If using hardwood or other charcoal, move over to 1 side of grill; if using gas, leave 1 section off.
Season chicken on both sides with salt and pepper, and drizzle well with canola oil. Place on the grill skin side down over direct heat, cooking until golden brown and slightly charred. Open and close the grill, as necessary, to maintain the heat without letting the skin burn. When skin is nicely charred and blistered, flip over and move to indirect heat, close lid and continue cooking until just cooked through, about 17 minutes longer, or until an instant-read thermometer inserted into the chicken registers 155 degrees F. Remove from the grill and let rest 10 minutes before cutting.
While the chicken is cooking, brush tomatoes, lemons and onion slices with oil, and season with salt and pepper. Grill lemons cut side down until golden brown, and tomatoes cut side down until slightly charred. Grill onion slices, flipping once, until softened and slightly charred.
Make the vinaigrette by whisking the vinegar and mustard until smooth. Slowly whisk in the olive oil, then whisk in the oregano and season with salt and pepper.
To serve, carve the chicken into 6 to 8 serving pieces and arrange in a large shallow serving bowl. Arrange the tomatoes and onions, all but 1 of the lemon halves, and the cucumbers over and around the chicken. Scatter the olives and feta cheese over, then drizzle with vinaigrette. Tear the basil leaves over the dish, and then dollop the yogurt in the center.
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