Spoon 1 tablespoon caper brine and 2 tablespoons drained capers into the bowl of a mini food processor. Add cilantro, 3 tablespoons oil, and 1 tablespoon water. Chop green onions to measure 3 tablespoons. Add 3 tablespoons green onions to cilantro mixture; process until smooth.

Cut remaining onions diagonally into 2-inch pieces. Cut 8 pieces lengthwise into slivers; set aside.

Heat a large skillet over medium-high heat. Sprinkle pork with ¼ teaspoon salt and pepper. Add remaining 1 tablespoon oil to pan; swirl to coat. Add pork and 2-inch onion pieces; cover and cook 4 minutes. Turn pork over.

Add tomatoes; cook 2 minutes. Sprinkle with remaining ⅛ teaspoon salt. Arrange pork mixture on a platter. Drizzle with sauce; sprinkle with slivered onions.