Brush steak with 1 teaspoon olive oil and sprinkle with herbes de Provence, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Heat ridged grill pan over medium-high heat. Add steak and cook until instant-read thermometer inserted into side of steak registers 145 degrees for medium, about 4 minutes per side.
Meanwhile, heat remaining 1 teaspoon oil in large skillet over medium-high heat. Add onion and cook, stirring, until softened, 4-5 minutes. Add mushrooms, thyme, remaining 1/4 teaspoon salt and remaining 1/4 teaspoon pepper. Cook, stirring frequently, until mushrooms brown, about 5 minutes. Add wine and cook, stirring, until mushrooms are tender and most of wine has evaporated, about 2 minutes longer.
Cut steak into 16 slices. Serve with mushrooms.
Serving Suggestion: green beans.
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