Heat oil over medium-high heat until just smoking.
Add mushrooms and cook, stirring occasionally, until beginning to brown and liquid has evaporated, about 5 minutes. Add shallot and cook, stirring often, until beginning to soften, about 1 minute. Increase heat to high; add red wine and broth.
SImmer rapidly until liquid and mushrooms are reduced to 1 cup, about 6 minutes.
Add vinegar and mustard; cook until thickened, about 1 minute.
Off heat, whisk in butter and thyme; season with salt and pepper. Spoon sauce over steaks and serve immediately.