Cut chayote halves lengthwise into 1/4-inch-thick slices and arrange in 1 layer in two thirds of a lightly oiled shallow baking pan. Put bell pepper quarters, skin sides up, in other third of pan. Roast in middle of oven, uncovered, until chayote is browned in spots and bell pepper skins are blistered, about 30 minutes. Immediately add chayote to dressing and toss to coat.