Bring a large pot of water to a boil.
While the water is boiling, melt 1 tablespoon butter in a 9-inch sauté pan set over medium heat. When the butter begins to sizzle and brown, add the chicken. Cook the chicken for 3 minutes on each side, or until golden brown. Remove chicken from the pan and set aside.
Return the pan to the burner and add the remaining butter. When the butter is melted, add the mushrooms, onions, and garlic. Cook for 5 minutes, or until mushrooms are golden brown and onion is soft. Sprinkle with flour and cook for one minute. Add the stock, half and half, rosemary, and thyme. Stir in the chicken. Simmer for 5 minutes, but do not allow it to boil.
Add the soba noodles to the boiling water and cook for 3 minutes (or as directed on the package). Drain and return to the pot.
Pour the sauce over the noodles and add the spinach. Stir to combine. Let sit for 2 or 3 minutes so the spinach can wilt. Season to taste with salt and pepper.