Pat the chicken dry, and sprinkle with ¾ teaspoon of salt, and ½ teaspoon of pepper. In a heavy, 12-inch skillet, heat the olive oil over medium-high heat. Add the chicken, skin-side down. Cover with a round of parchment, then place a heavy skillet on top, followed by a 3-5 pound weight (use a couple of 32 oz. cans). Cook for 10 minutes, then remove the weight, skillet, and parchment, and flip the chicken over. Cover again with parchment, skillet and weight. Cook about 8 minutes more, just until the chicken is cooked through. Transfer to a plate and cover to keep warm.
Add the squash, tomatoes, garlic, 2 teaspoons of the marjoram, and ¼ teaspoon of salt to the skillet. Cook over medium-high heat, stirring frequently, for 6-8 minutes, until the squash is tender and the tomatoes begin releasing their juices. If the chicken has released any juices on the plate, add these to the skillet and stir to combine. Season to taste with salt and pepper. Spoon the squash and tomato mixture over the chicken, and top with the remaining teaspoon of marjoram.