Arrange the chicken in a shallow baking dish, skin side up, and make 3 deep, crosswise cuts in each breast half. Season the chicken with salt and cover it with the puree, rubbing it into the slits. Spread the chayote, poblano and onion on a rimmed baking sheet, drizzle with oil and season with salt. Roast the chicken in the upper third of the oven for 30 minutes, basting a few times, until just white throughout; rotate the pan once halfway through. At the same time, roast the vegetables for about 30 minutes, until browned and the chayote is crisp-tender.