Add the beef to a large mixing bowl. Season all over with 1 teaspoon salt and a generous amount of black pepper. Toss the beef in with 2 tablespoons of the flour to coat.
Heat 2 tablespoons of olive oil in a large Dutch oven over medium heat. When the oil is hot, brown the beef on all sides, in batches, 4 to 6 minutes per batch. Remove the browned beef to a bowl and set aside.
Add 1 tablespoon of olive oil into the pot. Add the onions and carrots. Stir to coat the vegetables in the oil and season with 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the onion begins to wilt, 4 to 6 minutes. Add the tomato paste and thyme and stir to coat the vegetables. Add the remaining flour and stir to combine. Let the flour toast for a minute, then pour in the wine. Bring to a boil over medium heat and cook until slightly thickened and reduced, 3 to 5 minutes. Add the beef stock and bay leaves and adjust the heat so the sauce is at a moderate simmer. Return the beef and juices to the pot. Cover and reduce to a simmer, stirring occasionally, until the beef is fork tender, 2 to 2 1/2 hours, removing the lid in the last 15 minutes to reduce the sauce.
When the beef is almost done, heat a medium skillet over medium heat and add the remaining tablespoon of the olive oil. When the oil is hot, add the pearl onions, 1/4 teaspoon salt and a few grinds of pepper. Cook, tossing occasionally, until browned all over, 3 to 4 minutes. Add to the stew in the last 15 minutes of cooking,
Remove and discard the bay leaves (and thyme stems if using sprigs). Serve and garnish with parsley leaves and pita on the side.
In the bowl of a stand mixer fitted with a dough hook, mix together the water, yeast and sugar just to dissolve. Add the salt and oil, and then gradually add the flour. Increase the speed to medium-high and mix for 7 to 10 minutes, adding just enough additional flour so that the dough no longer sticks to the bowl. Do not add too much flour. The dough should be smooth and slightly sticky. Lightly coat a clean large bowl with oil or cooking spray and then place the dough in the bowl and turn it once or twice to coat it in oil. Cover the bowl with plastic wrap and let it rise at room temperature for 2 hours, or until it has doubled in size.
Turn the dough onto a clean work surface and divide it into 8 equal pieces. Mold each piece into a ball by stretching the top and tucking the edges under. place the balls 1-inch apart on a piece of parchment paper, cover them with plastic wrap, and let them rise for 30 minutes.
Set an oven rack in the middle position. Preheat the oven to 500 degrees F. Line two baking sheets with parchment.
With a rolling pin, roll out the balls of dough into 6-inch circles. Place them on the baking sheets and bake for about 5 minutes, or until they're puffy and just starting to brown.
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