In a large saucepan, heat 1 1/2 tablespoons EVOO over medium heat. Add the potatoes, carrot, celery, onion, bay leaf and thyme; season with salt and pepper, cover and cook for 10 minutes. Stir in the tomato paste. Make a well in the center of the vegetables, add the butter and cook until melted. Add the flour, stirring for 1 minute. Add the stock, bring to a simmer and cook until the potatoes are tender, 5 minutes. Discard the bay leaf.
In a large bowl, mix the beef with the parsley and Worcestershire sauce; season with salt and pepper. Form 4 patties, thinner at the centers for even cooking. In a large nonstick skillet, heat the remaining 1/2 tablespoon EVO over medium-high heat. Cook the patties, turning once, for 8 minutes for medium-rare.
While the burgers are cooking, toast and butter the english muffins.
On each of 4 plates, place a muffin half; top each with a patty and spoon some stew on top. Top with the cheese and chives.