Pat dry chicken breasts with paper towels and season them generously with 1 teaspoon salt and 1/2 teaspoon black pepper.
Heat butter and olive oil in a large skillet. Place chicken breasts and cook each side for 4 minutes, in medium high heat. At this point they will not be cooked thoroughly.
Place them on a plate, cover with aluminum foil and set aside.
Add the mushrooms, onions, and white wine to same skillet and cook for 8-10 minutes, stirring occasionally, until mushrooms are browned.
Add garlic and flour and give it a generous stir. Cook for 1 minute.
Add chicken broth; scrape the bottom to loosen the brown bits and cook until the broth is boiling.
Add the chicken and all the juices in the plate and cover it with a lid. Cook in medium heat, until the chicken registers 160 F. degrees when a thermometer is inserted in the middle. This will take 5 to 10 minutes.
In the meantime, whisk the egg yolk and sour cream in a small bowl.
Take the chicken breasts out and place it on a platter. Cover it with aluminum foil. Set aside.
Measure 1/2 cup of the mushroom sauce and stir it into the sour cream and egg mixture. Mix.
Stirring constantly, slowly pour the mixture into the skillet. Stir in the lemon juice, tarragon, and nutmeg. Let it simmer for 5 minutes.
Taste it for seasoning and add more, if necessary.
To serve, place a chicken breast on a plate and pour the sauce over it. Sprinkle it with more tarragon.