In a large pot, saute onion, carrot, and celery in olive oil and butter. Cook until onion starts to soften.: You will smell the vegetables sweetening as they release moisture, and the butter will melt into the oil creating a glossy sheen on the pan. The sound is a gentle sizzle, not an aggressive fry, indicating medium heat; visually the onion should turn translucent and the edges should not brown. This step matters because slow softening brings out the natural sugars that round the broth, avoiding any raw, vegetal bite. A common problem is using too high heat, which will brown or scorch the aromatics and give the soup an off flavor. If that happens, lower the heat and stir frequently to recover the gentle cook.
Stir in chicken stock and seasonings. Bring to boil and stir in chicken and noodles. Reduce the heat and cook until noodles are tender.: As the chicken stock hits the pot, it will steam and mingle with the sweet vegetable aromas, creating a fragrant uplift. When you bring the pot to a boil, you will hear a full, rolling bubble, but you should reduce to a gentle simmer before adding the chicken and egg noodles so they heat through more gently. The noodles will soften and swell, absorbing flavor; the chicken warms without drying out. This technique ensures the broth remains clear and the textures hold. Avoid a hard boil while the noodles cook because that motion breaks them up and causes cloudy broth. If the soup becomes too starchy, a splash of extra stock or a brief rest off heat helps redistribute the starch.
Taste, adjust seasonings, remove bay leaf, and serve.: The final stage is all about balance; take a spoon and sample the broth and noodle combination, tuning with salt and pepper to lift the flavors. Remove the bay leaf so no one bites into it, and give the soup a last whisk to distribute the aromatics. You should notice bright, herbaceous tarragon notes threading through the savory broth, and the noodles should be tender but not limp. A frequent oversight is underseasoning, which leaves the soup flat, so always taste and correct. If the seasoning seems off, add a pinch at a time and let it bloom for a minute before retasting.
