1. In a blender, puree oil, bell pepper, chile, onion, orange and lemon juices, cumin seeds, oregano, and 1 teaspoon salt until smooth. (Sauce can be made up to 3 days ahead and refrigerated; remove 30 minutes before serving.) Transfer to a serving bowl; just before serving, stir and top with more oregano leaves and a drizzle of oil.

2. Preheat grill for direct-heat cooking (medium-high on a gas grill). On a work surface, lay 3 steak pieces flat in a row. Insert a long metal skewer through all 3 pieces, about 1/2 inch from one short end. Insert a second skewer through all 3 pieces, 1/2 inch from other short end. To ensure even cooking, spread pieces 1/2 inch apart on skewers. Repeat with remaining 3 pieces steak. Pat meat dry and generously season both sides with salt; let stand 10 minutes. Lightly brush grates with oil. Grill steak, flipping once, until charred in places, 7 to 9 minutes for medium-rare; 10 to 12 minutes for medium. Let rest 10 minutes, then slice steak against the grain. Serve over rice, with pepper sauce, avocado, and more oregano leaves.