How to Make It
Process flour, eggs, egg yolk, and 1 teaspoon extra-virgin olive oil in a food processor until mixture is evenly moistened and a shaggy dough begins to form, about 15 seconds. Turn dough out onto a clean work surface; shape into a ball, and knead until smooth, about 3 minutes. Wrap dough tightly in plastic wrap; let stand at room temperature at least 1 hour or up to 4 hours.
Unwrap dough; cut in half. Using your hands, flatten 1 dough portion to 1/3-inch thickness, dusting with 00 flour as needed to prevent sticking (keep other dough portion covered with a towel). Roll flattened dough portion through pasta machine with rollers on widest setting. Fold dough in half crosswise, and reroll through pasta machine (keep rollers on widest setting). Continue rolling dough through pasta machine, reducing width of rollers 1 setting at a time, until dough has been rolled through second-thinnest setting, flouring as needed to prevent sticking. Repeat process with remaining dough portion.
Place pasta sheets on a work surface dusted with semolina flour. If desired, trim edges of pasta sheets using a fluted pastry wheel. Cut pasta into irregular 3-inch triangles or squares. Place pasta pieces on a semolina-dusted rimmed baking sheet, lightly dusting tops of pasta with semolina between layers. Cover tightly with plastic wrap; chill until ready to use, up to 4 hours.
Sprinkle chicken pieces all over with pepper and 2 teaspoons salt. Heat olive oil in a Dutch oven over medium-high. Working in batches if needed, add chicken, and cook until browned on both sides, 6 to 8 minutes. Transfer chicken to a plate; set aside.
Add onion to chicken drippings in Dutch oven. Cook over medium-high, stirring occasionally, until onion is softened and golden brown, 10 to 14 minutes. Add wine; cook, stirring occasionally to loosen browned bits on bottom of pan, until liquid is reduced by half, about 2 minutes.
Return chicken to Dutch oven; add broth and rosemary sprigs. Bring to a simmer over medium-high. Reduce heat to low; cover and cook until a thermometer inserted in thickest portion of chicken thighs registers 165°F, 20 to 25 minutes. Remove and discard rosemary sprigs. Transfer chicken to a plate; let stand until cool enough to handle, about 10 minutes.
Meanwhile, continue cooking mixture in Dutch oven over low, uncovered, until slightly thickened and reduced to about 13/4 cups, 14 to 16 minutes.
Shred chicken meat; discard bones and skin. Stir shredded chicken into mixture in Dutch oven. Season with salt and pepper to taste. Cover and keep warm over low.
Bring 3 quarts water to a boil in a large saucepan over medium-high. Stir in remaining 3 tablespoons salt; add pasta. Cook, stirring occasionally, until tender, 3 to 4 minutes. Drain. Add pasta to chicken mixture; toss to coat. Drizzle with remaining 1 tablespoon extra-virgin olive oil. Serve immediately.