Over medium heat in a large saucepan, heat one tablespoon of neutral oil until shimmering. Add your chicken and sear on both sides until deeply golden brown, about 3-5 minutes per side. Remove from the pan and set aside.

Reduce the heat to medium-low and melt butter. Add in onion and a pinch of salt and cook, stirring occasionally, until softened, scraping off any browned bits from the bottom of the pan as the onion releases moisture, about five minutes. Add garlic and cook until very fragrant, about 30 seconds.

Sprinkle flour over onion and garlic mixture and cook, stirring constantly, until the raw flour smell has cooked off, about 1-2 minutes. Return the chicken to the pot (along with any accumulated juices) and pour the stock over it. Bring the soup to a boil, reduce the heat to a simmer, cover, and allow to simmer for about 1 hour.

Uncover and remove the chicken from the soup. Set aside. Lightly beat the egg yolks in a bowl. Whisking constantly, slowly ladle a small amount of the hot broth into the egg yolks to prevent them from curdling. Pour the yolk mixture into the pot and stir to combine.