Season the rib-eye cutlets all over with 2 tsp. salt and 1 tsp. pepper. Heat a large, well-seasoned cast-iron skillet over medium-high heat for 2 to 3 minutes. Add 1 Tbs. of the oil, arrange two of the cutlets in the skillet, and cook, flipping once, until seared and golden-brown on both sides, about 2 minutes total. Transfer to a large plate and keep warm. Repeat with the remaining four cutlets, adding an additional 1 Tbs. oil before the last batch.
Reduce the heat to medium low. Add the breadcrumbs, almonds, and pepper flakes to the skillet and cook, stirring constantly and scraping up any browned bits, until the mixture is fragrant and toasted, about 2 minutes. Add the peppers, garlic, tomato, 1/2 tsp. salt, and 1/8 tsp. pepper and cook, stirring often, until softened and juicy and the flavors have melded, 3 to 4 minutes.
Transfer the mixture to a food processor. Add the vinegar and 2 Tbs. warm water and purée until smooth. With the motor running, drizzle in 2 Tbs. of the oil. Transfer the sauce to a bowl, season to taste with salt and pepper, and keep warm.
Heat the remaining 1 Tbs. oil in the skillet over medium heat. Add the shallots and cook, stirring often, until just softened, 2 to 3 minutes. Add the spinach, cover, and cook for 1 minute. Uncover and continue to cook, tossing often, until just wilted, about 2 minutes more. Remove from the heat and toss in the olives, lemon juice, orange juice, 3/4 tsp. salt, and 1/8 tsp. pepper.
Spoon the romesco sauce onto 6 plates, top with the cutlets and wilted spinach, and serve.