Melt butter in a skillet until sizzling; stir in onion, peppers and garlic. Cook over medium-high heat 2 to 3 minutes or until peppers are crisp-tender. Remove vegetables from skillet with a slotted spoon and set aside.
Add chicken, tarragon, salt and pepper to skillet. Continue cooking, stirring occasionally, 7 to 9 minutes or until chicken is golden and tender.
Add vegetables, half-and-half and cheeses to chicken mixture. Reduce heat to medium; continue cooking 3 to 5 minutes or until cheese is melted. Add vermicelli; toss gently to coat. Serve immediately.