Place lentils and 2 cups broth in a small saucepan and bring to a boil. Boil 5 minutes, then reduce to a simmer and cook until lentils are tender and most liquid is dissolved, about 30 minutes. Drain any remaining liquid. While lentils are cooking, place bulgur and remaining 2 cups chicken broth in another small saucepan and bring to a boil. Reduce heat and simmer until most liquid is dissolved and bulgur is tender, about 13 to15 minutes. Remove from heat and fluff with a fork. Add the bulgur to the lentils.
Heat 2 teaspoons of the oil in a skillet over medium heat. Add onions and cook, stirring, until tender, about 5 minutes. Add peppers and cook until peppers are tender, another 5 minutes. Add the onion mixture to bulgur-lentil mixture. Stir in parsley, basil, chives, lemon zest, lemon juice remaining 1 tablespoon olive oil, salt and pepper and toss to incorporate.