Season the chicken generously with salt and pepper. In a large pan with straight sides and a lid (large enough to eventually hold all the chicken nestled in 1 layer), add the olive oil over medium-high heat. Once the oil begins to swirl, add the chicken pieces in batches, without crowding the pan. Cook until golden brown, flipping and browning on all sides. The purpose isn't to fully cook them here, just to get color. Remove the chicken to a plate.
Immediately add the tomatoes to the pan and allow them to roast while resting for a few minutes, then shake the pan and stir to get another side of the tomatoes roasting on the bottom of the pan. Cook like this until the tomatoes are charred and begin to break apart and create a sauce, about 8 minutes.
Add the black olives, chile flakes and garlic to the tomatoes. Then, nestle the chicken pieces back into the pan. Add enough chicken stock so the chicken is almost but not completely covered by the liquid (leave about 1/2 inch). Cover and lower to a simmer. Cook until the check of a garlic clove reveals it to be tender, or the chicken is cooked through, about 30 minutes.
Bring the liquid in the pan back to a boil. Add the pasta and spinach and cook until the pasta is al dente, about 5 minutes (according to the timing on the package). Remove the pasta to serving plates or bowls. Pour the chicken and spinach sauce over the pasta and serve with a spritz of lemon.