01 - Mound flour on a clean work surface and create a deep well in the center. Crack eggs into the well and add salt. Using a fork, gradually incorporate flour into eggs until a shaggy dough forms. Knead by hand for 8-10 minutes until smooth and elastic. Wrap in plastic and let rest at room temperature for 30 minutes.
02 - Heat olive oil in a skillet over medium heat. Add chopped onion and sauté until softened, about 3 minutes. Add minced garlic and mushrooms; cook until mushrooms release their moisture and turn golden, approximately 5 minutes. Remove from heat and stir in shredded chicken, grated Parmesan, parsley, black pepper, and salt to taste. Allow filling to cool completely.
03 - Divide rested dough into two equal portions. Working with one piece at a time, roll dough into thin sheets using a pasta machine or rolling pin until approximately 1/16 inch thick. Keep remaining dough covered to prevent drying.
04 - Place small mounds of filling (about 1 tsp each) onto one pasta sheet, spacing them 1 inch apart. Lightly brush edges with water and carefully place second pasta sheet over filling. Press firmly around each mound to seal. Cut into individual squares and crimp edges with a fork to ensure tight seal.
05 - Heat olive oil in a saucepan over medium heat. Add sliced garlic and sauté until fragrant, about 1 minute. Pour in crushed tomatoes and add sugar, salt, black pepper, and oregano. Simmer uncovered for 15-20 minutes until slightly thickened. Stir in torn basil leaves just before serving.
06 - Bring a large pot of salted water to a rolling boil. Gently lower ravioli into water and cook for 3-4 minutes, or until they float to the surface. Remove carefully with a slotted spoon, allowing excess water to drain.
07 - Divide cooked ravioli among serving plates. Spoon warm tomato sauce generously over pasta. Garnish with additional grated Parmesan cheese and fresh basil leaves. Serve immediately while hot.