Cut each chicken breast in half on the diagonal, making each piece about the same weight. Season with salt then rub with 2 tablespoons olive oil.
Using a blender or food processor, purée the tomatoes, basil, almonds, garlic, 1 to 2 teaspoons salt and the crushed red pepper for about 1 minute. Scrape down the bowl. With the machine on, pour in the remaining oil in a steady stream to emulsify.
Preheat a large cast-iron or nonstick skillet. Working in batches if necessary to avoid overcrowding, add the chicken breasts and sear until golden, a few minutes on each side (make sure not to overcook).
Transfer the chicken pieces to plates and top with the pesto. To serve, sprinkle the cheese on top.