In a skillet melt 3 tablespoons of the butter over medium heat and cook the mushrooms stirring occasionally, until slightly gold at the edges, about 3 minutes. Stir in 1 more tablespoon of the butter, the shallots, garlic and salt. Reduce the heat to low and cook, stirring occasionally, until the shallots are softer and translucent, about 5 minutes.
Sprinkle the flour over the shallots – mushrooms and cook for about another minute stirring well and incorporating the flour. Add the wine, stock, thyme springs and salt. Raise the heat and simmer until the liquid is reduced by half (1 cup), about 5 minutes. Pass the mixture through a fine strainer pushing firmly on the vegetables until all the liquid has been extracted. Discard the solids and set the wine mixture aside.
Generously salt and pepper both sides of the steaks.
In a large skillet heat the oil over medium high heat. When hot add the steaks and cook until dark brown and crusty, 4 – 6 minutes per side for medium rare. Transfer to a plate, cover with aluminum foil and let them rest.
Let the skillet cool off, remove any fat or oil remaining in the skillet and do not wipe it. Add the reserved wine mixture and bring to a low simmer, scraping the bottom of the skillet to loosen any browned bits. Simmer until reduced by a one-forth.
Reduce the heat and add the remaining tablespoon of butter. Add any juices accumulated from the steaks. Season with salt and pepper if needed.
Divide the steaks between plates and pour the sauce over it.