For the soup: Heat the olive oil in a large Dutch oven over medium heat. Add the onions, reduce the heat to low, and cook, stirring occasionally, until deep golden brown, about 1 hour and 30 minutes.
Stir in the paprika, peppers, tomatoes, beef and salt. Reduce the heat to low, cover, and cook, stirring occasionally, until most of the liquid at bottom of the pot has evaporated, about 3 hours.
Add the carrots, parsley root, celery and caraway seeds, and cook, covered, stirring occasionally, until the vegetables have softened, about 30 minutes.
Make the "pinchie" noodles: Using your hands, mix the eggs, flour and a pinch of salt in a medium bowl until the dough begins to pull away from sides of bowl. Form the dough into a ball and cover with a damp towel.
Add the beef broth and potatoes to the pot and bring to a boil. Pinch pea-sized pieces of dough into the soup until all of the dough is used. Cook until the noodles firm up and the potatoes are tender, about 5 more minutes.
Remove the soup from the heat and serve immediately with slices of rye bread.
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