Preheat the oven to 350 degrees F.
Generously salt and pepper your eye of round, then sprinkle with your paprika and oregano. Heat the olive oil in a large Dutch oven over medium heat. Add the eye of round and brown all over, 4 to 5 minutes total. Remove to a plate.
Add the garlic, bell peppers, onions and thyme to the Dutch oven and cook until the vegetables just begin to brown, about 2 minutes. Add the meat back in and add the wine, then increase the heat to boil the wine down to a glaze, 1 to 2 minutes. Add the broth and reduce the heat to low so that the liquid is simmering.
Cover and place in your preheated oven and continue to cook until the meat gently falls apart, about 2 ½ to 3 hours.