Position a shelf at the lowest position in the oven and preheat to 450 degrees F.
Cut the backbone out the chickens with kitchen scissors and reserve for another use or discard. In a shallow dish combine the thyme, 9 cloves crushed garlic, lemon juice and 1/3 cup of extra-virgin olive oil. Add the chickens and coat thoroughly in the mixture, cover and marinate in the refrigerator for up to 1 hour.
Remove the chicken from the marinade, discard marinade. Turn the birds breast-side up and fold the wing tips back and under. Season both sides with some salt and pepper and the paprika. Place the chickens bone-side up on a broiler pan and position the legs in the center. Roast in the oven until the chickens are 3/4 cooked (about 140 to 150), about 25 to 30 minutes.
Put the whole grape tomatoes, scallions, garlic cloves, and basil leaves in a large bowl, drizzle with 1 tablespoon of the olive oil and sprinkle with salt and pepper to taste. Turn the vegetables until they are lightly coated. After the chickens have cooked for about 25 minutes, arrange the vegetables around them. Turn the oven to broil and broil until the vegetables are charred, about 10 to 15 minutes. An instant-read thermometer should reach 170 degrees F. Transfer the chicken and vegetables to a cutting board. Pour off all except about 2 tablespoons of chicken fat from the pan and set aside.
Meanwhile, bring a large pot of cold water to a boil and salt it generously. Add the spaghetti, and cook, stirring occasionally, until al dente, 8 to 10 minutes.
Drain, reserving 1/2 to 3/4 cup of the pasta water and transfer the both to a serving bowl.
Squeeze the garlic cloves from their skins, and chop them up with the scallions, once chopped transfer to a bowl and add the charred grape tomatoes, stir to make a chunky relish.
Once the chicken is cooked and resting, place the pan you broiled the chicken in on the cook top over medium heat. Add the red wine vinegar, Worcestershire sauce and the reserved pasta cooking water, bring up to a simmer scraping the bottom of the pan with a wooden spoon. Add the cooked spaghetti and chopped parsley, stir to coat and continue to cook for 1 minute. Transfer the spaghetti to a serving bowl.
Carve the chicken, top with some of the chunky relish and serve with some of the spaghetti on the side. Garnish with grated Parmesan, if using.
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For cooking the chicken: .