Cut the chicken breasts into 3 equal pieces (in half crosswise, then the thicker half again in half, lengthwise). You can also pound it lightly to an even thickness (it will cook evenly and be juicy. Season to taste with salt and pepper.
Cut the butternut squash into very small cubes, about 0.5 cm / 1/4 inch. You can also grate it on the big holes of the grater, but I think it looks more appetizing when you cut it in cubes. If you're using pumpkin puree add it later along with the broth and cream.
Cut the mushrooms into thick, 0.5 mm / 1/4 inch slices.
Chop finely the shallots (or onion) and garlic.
Heat the oil in a large frying pan over high heat, add in the chicken pieces. Cook on one side for about 3 minutes, until golden, turn over and cook for a minute. Be careful not to overcook the meat, or it will be tough. Remove from the pan, it should be slightly pink in the middle (it will be cooked through later in the sauce) or no longer pink. Transfer on a plate.
Add the mushrooms and butternut squash to the pan, spread them evenly. Cook over high heat for about 3 minutes, don’t stir (!). When they are sightly browned at the bottom - stir, then add the onion with garlic. Reduce the heat to medium-low. Cook, stirring from time to time for about 4-5 minutes.
Pour in the cream and broth (and pumpkin puree if using), season with nutmeg and cook over medium-low heat for a few minutes, until the sauce thickens a bit. Don‘t worry if the sour cream breaks into small clumps – the sauce will be smooth again after cooking for a few minutes.
Add the chicken back to the pan, cook until cooked through.
Remove from the heat, season with salt and pepper, stir in chopped dill.
Serve with potatoes and optionally with other vegetables, like cooked green beans.
Enjoy!