Cut the potatoes into 1-2 inch chunks, cover them with water in a medium pot and bring to a boil. Simmer until the potatoes are tender, about 15-20 minutes. Drain and return the potatoes to the pot.
Add the butter and mash the potatoes with a potato masher. Add the milk and mash again. Season to taste with salt and pepper and set aside.
Heat the olive oil in a large deep skillet over medium high heat. Add the onions and carrots and saute until the onions are tender, about 5 minutes. Add the Worcestershire sauce and the beef and stir until the beef is browned and cooked through. Stir in the tomato paste and stir for another minute or two.
Assemble the quesadillas as follows: Lay four tortillas out on the counter. Divide the mashed potatoes among the four tortillas and spread the potatoes to cover each tortilla. Divide the beef mixture among the tortillas and spread it to the edges. Sprinkle with cheese and top with another tortilla.
Heat a clean large skillet over medium high heat. Place each quesadilla one by one into the pan and brown both sides, flipping them carefully when the first side is done.
When all the quesadillas are browned, cut them into wedges (I use a pizza cutter for this) and serve at once.