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