Heat oil and butter in a soup pot, then sweat carrots, onion, celery, and garlic with salt for 8-10 minutes. Add chicken stock, bring to a boil, then simmer for 30 minutes.

Whisk salt and flour, then add eggs to a well in the center and stir until a sticky dough forms. Knead the dough on a floured surface until smooth and dense, then let it rest for 20 minutes.

Divide dough into four balls. Roll out each ball thinly, dust with flour, and roll up like a sleeping bag. Cut into 1/4 inch or 3/4 inch slices, unroll, and set aside.

Bring soup to a boil, then add noodles gradually, stirring to prevent sticking. Incorporate parsley and chicken, cook for 15 minutes, adjust seasonings, stir in lemon juice, and serve.