Place poblano peppers on a foil-lined baking sheet; broil 8 minutes. Wrap in foil; let stand 10 minutes. Peel and coarsely chop. Combine peppers, tomatoes, onion, 2 tablespoons oil, juice, and salt in a bowl. Combine potatoes, oats, red pepper, flour, and egg in a bowl.
Heat 1 1/2 tablespoons oil in a large skillet over medium-high. Spoon about 1/3 cup potato mixture into pan. Repeat 3 times to form 4 cakes. Cook 3 to 4 minutes on each side. Remove from pan; keep warm. Repeat procedure with remaining 1 1/2 tablespoons oil and potato mixture. Top cakes evenly with poblano mixture, chicken, and cilantro.