- For pasta, place flour in a large bowl and make a well in the center.
- In a large measuring pitcher, combine egg yolks and olive oil. Whisk well.
- Pour yolk mixture into the well. Using a fork, slowly incorporate flour into liquid. Once dough starts to come together, pour it from bowl onto a clean work surface and knead with your hands until the dough is smooth and has nice elasticity (it should spring back immediately when you press it), about 10 minutes. Cover with plastic wrap and let pasta dough rest for 45 minutes.
- To prepare filling, place chicken thighs and ham in the bottom of a Dutch oven. Add garlic, white wine, cream, and stock. Turn heat to medium and bring liquid to a boil. After boiling, reduce heat to a small simmer and close lid until thighs are cooked through and fork tender, approximately 45 minutes.
- Once thighs are cooked, strain cooking liquid and reserve.
- Pick the chicken meat from the bones.
- In a food processor, add picked thigh meat, Pecorino Romano, eggs, parsley, salt and pepper. Purée until smooth, adding cooking liquid in ¼ cup at a time as needed to keep filling moist (but not wet or loose). Allow filling to cool, then place in a piping bag and keep cool until ready to use.
- After pasta dough has rested, cut into 8 pieces.
Note: Cover unused portions of dough with a damp kitchen towel as you work.
- To make pasta sheets, working with 1 piece at a time, roll pasta dough through a standard pasta machine to the number 5 setting.
- When sheets are about as thin as a piece of paper, cut rounds using a 1-inch round cutter.
- Pipe about 3 teaspoons of filling into center of a pasta round, being careful not to add too much. Dip your finger in water and coat the edge of half of the pasta round. Fold the other half of the pasta round over filling, lining up pasta edges to make a half moon.
- Place the filled half moon on work surface with round edge facing away. Place a finger of your nondominant hand in the center of the half moon, then use your other hand to bring the two sides together over your finger. Pinch sides together where they connect to seal the cappalletti. Repeat with remaining pasta rounds and filling.
- Bring a pot of well-salted water to a rolling boil. Add pasta and cook until just al dente, and drain, reserving pasta water.
- To prepare the sauce and serve, add butter in a separate pan over medium heat, and melt. As butter begins to foam, add sage, carefully watching so as not to brown the butter, about 2 minutes. Just before butter browns, add about 4 ounces of pasta water, then pasta. Season with salt and pepper. As sauce thickens, spoon cappallettis onto each plate and finish with cheese and freshly cracked pepper.