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 to 10 minutes until smooth and elastic. Wrap tightly in plastic wrap 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 chopped mushrooms; cook until mushrooms release moisture and become golden, approximately 5 minutes. Remove from heat and let cool slightly. Stir in shredded chicken, grated Parmesan, chopped parsley, black pepper, and salt to taste. Allow filling to cool completely.
03 - Cut rested dough in half and work with one portion at a time, keeping the other covered. Roll dough into thin sheets using a pasta machine or rolling pin until approximately 1/16 inch thick. If using a pasta machine, progress to the second-to-thinnest setting for optimal results.
04 - Place rounded teaspoons of filling 1 inch apart on one pasta sheet. Lightly brush water around each mound of filling. Carefully place second pasta sheet over the first, pressing down around filling pockets to expel air. Cut into individual squares using a knife or ravioli cutter. Seal edges firmly by pressing with fork tines to prevent opening during cooking.
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 stir in sugar, salt, black pepper, and dried oregano. Reduce heat to low and simmer for 15 to 20 minutes until slightly thickened. Remove from heat and fold in torn fresh basil leaves.
06 - Bring a large pot of generously salted water to a rolling boil. Gently drop ravioli into boiling water, being careful not to overcrowd. Cook for 3 to 4 minutes until ravioli float to the surface. Remove carefully with a slotted spoon to avoid tearing.
07 - Transfer cooked ravioli to serving plates or shallow bowls. Ladle warm tomato sauce generously over the pasta. Sprinkle with additional grated Parmesan cheese and garnish with fresh basil leaves. Serve immediately while hot.