In a large skillet, sauté onions and carrots about 3 minutes. Add ground beef, cooking and stirring until beef has browned. Drain off fat. Add broth, tomato paste, and herbs. Cook and stir until sauce has reduced and thickened slightly. Stir in the peas. Pour the beef mixture into a 1 1/2 quart baking dish and set aside.
Meanwhile, put the potatoes in a medium pot and cover with water. Bring to a boil and cook until potatoes are fork-tender, about 20 minutes. Drain. Mash the potatoes in the pot with butter and milk.
Spread mashed potatoes over beef mixture. Bake in preheated 375 degree F oven for 30 to 35 minutes, until potatoes turn a light golden brown around the edges.