How to Make It
Combine 6 cups water, chicken thigh, carrot, and celery in a large saucepan. Bring to a low simmer (150°F to 160°F) over medium-low, and cook 10 minutes. Add pork and beef. Cook until an instant-read thermometer inserted into thickest part of chicken registers 165°F, about 5 minutes. Remove meat from pan, and let stand until cool, about 15 minutes. Discard poaching liquid and vegetables.
Pick chicken meat off bones, discarding bones and skin, and place in a food processor. Add beef and pork; pulse until finely minced but not pasty, 12 to 16 times. Transfer meat mixture to a medium bowl. Add 1 1/4 cups Parmigiano-Reggiano cheese, salt, and nutmeg. Stir together until very well incorporated; taste for seasoning, and adjust if needed. (The filling should be highly seasoned.) Stir in egg. Cover and chill until ready to fill tortellini, up to 6 hours.
Mound flour on a wooden board, and make a well in center. Crack eggs into well, and add olive oil. Using a fork, beat together eggs and olive oil, and slowly incorporate flour into egg mixture until a dough forms. Using your hands, bring dough together, kneading until smooth and elastic, about 10 minutes. Wrap dough tightly in plastic wrap, and set aside to rest at room temperature 1 hour or up to 3 hours.
Unwrap dough, and divide into 4 portions. Working with 1 dough portion at a time (keeping other portions covered with a towel), flatten dough to 1/3 inch thickness. Using a pasta machine set to widest setting, roll flattened dough through machine. Fold dough in half crosswise, and reroll through pasta machine. Continue folding and rerolling dough until it is at least 4 inches wide, 1 or 2 more times. Once appropriate width is reached, continue rerolling dough through pasta machine, reducing width of rollers 1 setting at a time, until dough has been rolled through setting 0 (narrowest setting), sprinkling dough with flour as needed to prevent sticking.
Cut 1 dough sheet into about 28 (2-inch) squares. Place 1/4 teaspoon filling in center of each square; brush edges of pasta lightly with water. Fold dough in half over filling, forming a triangle shape. Gently press out all of the air around filling. Bring up 2 corners of the triangle so that they meet at the bottom; press points to seal together, making a traditional tortellini shape. Place shaped tortellini in a single layer on a parchment paper–lined baking sheet. (Keep tortellini covered with damp paper towels as you make them.) Repeat with remaining dough sheets and filling. Refrigerate until ready to cook.
Bring broth to a boil in a large saucepan over medium-high. Working in batches, add tortellini, and cook until tender, 6 to 8 minutes. Divide tortellini evenly among shallow bowls, and top each with 1/3 cup or more warm broth. Sprinkle bowls with remaining 1 cup Parmigiano-Reggiano, and serve immediately.