Heat oven to 350F. Butter a small baking dish (about 8 inches in diameter.
Cook potatoes and carrots in boiling, salted water until bust tender.
Arrange a layer of potatoes and carrots in the baking dish, sprinkle lightly with salt and pepper, sprinkle with a tablespoon of herbs, sprinkle with a third of the cheese. Repeat for two more layers. Add chicken broth.
Cover tightly with foil and bake for 15 minutes. Remove foil and cook another 15 minutes.