Peel onions and chop into coarse pieces. Melt lard in a heavy 6 to 8-quart Dutch oven. Add the beef cubes to the oil and brown. Work in batches if necessary, removing cubes as they are browned. Don't overcrowd the pan. Add onions to the pot. Heat should be low in order not to brown the onions. When onions become glossy, add back the seared beef. Stir.
Meanwhile, chop and crush the garlic with the caraway seeds and a little salt; use the flat side of a heavy knife.
Take kettle from heat. Stir in paprika and the garlic mixture. Stir rapidly with a wooden spoon. Immediately after paprika is absorbed, add 2 1/2 quarts warm water. (Cool water toughens meat if you add it with the meat is frying.)
Replace covered kettle over low heat and cook for about 1 hour.
While the braising is going on, peel the tomato, then cut into 1-inch pieces. Core green peppers and slice into rings. Peel potatoes and cut into 3/4-inch dice.
After the meat has been braised for about 1 hour (the time depends on the cut of the meat), add the tomato and green peppers and enough water to give a soup consistency. Add a little salt. Simmer slowly for another 30 minutes.
Add potatoes and cook the gulyas till done. Adjust salt. Add hot cherry pepper pods if you want to make the stew spicy hot.
Cook the dumplings in the stew.
Serve the gulyas steaming hot in large, extra-deep bowls. The meat should be tender, but not falling apart.
Mix the egg with flour and salt.
Spoon the mixture into boiling soup, using 1/4 teaspoon at a time. Cook in the soup for 2 or 3 minutes just before serving.
Recipe courtesy George Lang, Cuisine of Hungary, Random House Value Publishing, February 25, 1990