Heat the olive oil in the Dutch oven until shimmering.: The aroma of hot olive oil will be bright and slightly fatty when the surface begins to ripple, and you may see faint wisps of heat rising from the pan. This shimmering is the cue that the oil is ready to accept aromatics without absorbing them too quickly, which helps prevent soggy onion . If the oil smokes heavily, your pan is too hot, so reduce the heat briefly to avoid burning and a bitter taste.
Add onion and saute until starting to soften (2 to 3 minutes).: You should hear a gentle sizzle as the onion hits the oil, and within a couple of minutes the pieces will turn translucent and give off a sweet, fragrant scent. Softening rather than browning preserves delicate sweetness which supports the broth. A common mistake is overcrowding the pan, which cools it down and causes steaming instead of sauteing, so give the onions room to move.
Add mushrooms and cook uncovered, stirring often, until soft (about 4 minutes).: As the mushrooms hit the pan they will release moisture then begin to soften and take on a deeper color, giving off a roasted, earthy aroma. Stirring often encourages even evaporation so they brown slightly rather than steam. Avoid leaving them untouched for too long, which can cause uneven cooking where some pieces overcook while others remain raw.
Stir in the chicken broth, put on the lid and bring to a boil.: When you add the chicken broth the pan will go from hot and dry to lively and aromatic, releasing a savory steam. Put the lid on to trap heat and speed the transition to a boil, which concentrates the flavors and prepares the liquid to cook the noodles. If the broth tastes flat, a pinch of salt can brighten it, but add cautiously since store broths vary in sodium.
Stir in the egg noodles. Put the lid back on and cook the soup, stirring occasionally, for 6 minutes.: The moment the egg noodles meet the bubbling broth they will start to hydrate and release a faint wheaty scent as they plump. Give them a quick stir so they separate and do not stick together. A helpful tip is to use a wide spoon to make sure none clump at the bottom, as clumped noodles cook unevenly and can stay gummy in the center.
Stir in the paprika and the chicken. Continue to cook the soup until the noodles are soft (about another 2 to 3 minutes).: With the lid on the pot, the broth will maintain a steady simmer and the egg noodles will absorb liquid, swelling and softening. You will notice the surface quiet down and the noodles rising nearer the top as they cook. Stirring occasionally prevents sticking and helps you judge doneness, since undercooked noodles will be dense, while overcooked ones become mushy.
Turn the heat off and let the soup cool for 1 minute, then stir in the sour cream and serve.: When you add the smoked paprika it should bloom in the warm broth releasing its smoky perfume, and the shredded chicken will warm through quickly, contributing savory, meaty notes. The broth should take on a faint rosy hue from the spice. A misstep here is adding dairy before cooling slightly, which can cause curdling; keep the heat moderate and stir gently.
Continue to cook the soup until the noodles are soft (about another 2 to 3 minutes): In these final minutes the noodles finish absorbing flavor and the chicken melds with the broth. You should see the surface develop small, steady bubbles and smell an integrated savory aroma. Taste a noodle for texture, aiming for tender with a slight bite. Overcooking will make the noodles too fragile, causing them to disintegrate when stirred.
Turn the heat off and let the soup cool for 1 minute, then stir in the sour cream and serve: Allowing the pot to rest for a minute cools the liquid enough to prevent the sour cream from breaking, and when stirred in it transforms the broth into a silky, lush finish. You will notice a creamy sheen and a softer mouthfeel immediately. A common error is to add the sour cream directly to boiling soup, which can cause separation, so always temper this step by removing from heat first.