Bistro-Style Filet Mignon with Champagne Pan Sauce

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4 beef tenderloin steaks (filet mignon), cut 1 inch thick (about 6 ounces each)

1/ teaspoon coarse grind black pepper

Champagne Pan Sauce:

1 tablespoon olive oil

1-1/2 cups assorted mushrooms, such as shiitake, cremini or button, sliced

2 cloves garlic, minced

3/4 cup reduced sodium beef broth

1/2 cup brut Champagne or sparkling wine

1/2 teaspoon chopped fresh thyme or 1/4 teaspoon dried thyme

1 teaspoon cornstarch

1 tablespoon water