In a large bowl, combine 3 tablespoons olive oil, chile powder and oregano. Mix to blend, then pour half of the mixture into another large bowl. Cut the chicken crosswise into 1/2-inch wide slices, and then cut the large pieces in half lengthwise and crosswise to create slices of about the same size. Add to one of the bowls, sprinkle with salt and pepper and toss to coat.

Cut the onions in half through the stem end, then peel off the dried outer layers. Place the onions cut side down on work surface. Cut lengthwise into ¼-inch-thick slices. Add to the second bowl. Sprinkle with salt and pepper and toss to coat.

Heat a heavy large skillet over medium-high heat. Add enough oil to coat with a thin layer. Add the onions and cook until they soften and begin to brown on the edges, turning frequently, about 6 minutes. Transfer to a platter. Rinse out the skillet and dry. Place over medium-high heat. Add enough oil to coat with a thin layer. Add the chicken and cook, turning occasionally, until cooked through, about 5 minutes. Taste and adjust seasoning with salt and pepper. Arrange the chicken over the onions. Garnish with cilantro and serve with lime wedges and tortillas.