Steak and Eggs with Saucy Beans

Name: Steak and Eggs with Saucy Beans
Ingredients: 2 tsp. hot smoked Spanish paprika; 2 tsp. Aleppo-style pepper; 2½ tsp. kosher salt, plus more; 1 tsp. freshly ground black pepper, plus more; 2 1¼-thick boneless New York strip steaks (about 14 oz. each), patted dry; 7 Tbsp. extra-virgin olive oil, divided; 2 medium shallots, finely chopped; 4 garlic cloves, thinly sliced; 1 small bunch cilantro, stems finely chopped, leaves coarsely chopped; 1 15.5-oz. can pinto beans, drained; 2 Tbsp. unsalted butter; 3 limes, halved; 4 large eggs