Coat a large, wide, straight-sided pan with olive oil and bring to high heat. Season the chicken generously with salt and add it, skin side down, to the pan—you should hear a big sizzle. If you don’t, remove the chicken and wait. When you put chicken in a hot pan, the first thing it wants to do is stick there and the first thing you want to do is move it. Resist the urge. It will unstick itself when it’s ready. When the skin is brown and crispy, 5-7 minutes, turn the chicken over and brown the other side. Remove the chicken from the pan and reserve.
Ditch the fat and lower the heat. Add another splash of olive oil to the pan and add the onions. Season with salt and red pepper and cook over medium heat for 8-10 minutes or until the onions are soft and aromatic; then add the garlic and cook for another 2-3 minutes, stirring frequently.
Add the mushrooms, season with salt, and cook for 4 to 5 minutes or until soft and aromatic. Add the wine and cook until reduced by half, 3 to 4 minutes.
Return the chicken to the pan, pour in enough stock to almost cover the chicken, and add the thyme bundle and bay leaves. Bring to a boil (BTB), reduce to a simmer (RTS), and cook for 30 minutes. Add a little more stock if the liquid level gets low.
While the chicken cooks, puree the almonds in a food processor. Once they are ground, drizzle in a little olive oil while the machine is running to make a loose paste. Season with salt and reserve.
When the chicken has simmered for 30 minutes, remove it from the pan and reserve; remove the bay leaves and thyme bundle and discard. Stir the almond puree into the sauce and taste for seasoning, adding more salt if needed. Bring to a boil (BTB) and reduce to a simmer (RTS) to thicken the sauce if needed. Serve the chicken draped with the sauce and garnished with chives.