1. Prepare a medium-hot fire in a charcoal grill, or heat a gas grill to high. Brush steaks with oil; season with salt and pepper.
2. Grill steaks until cooked to desired doneness, 5-6 minutes per side for medium-rare. Transfer steaks to a plate and let rest for 5 minutes
3. Meanwhile, heat 1 tablespoon oil in a large cast-iron skillet set on grill alongside steaks. Add mushrooms and shallots to skillet and stir until mushrooms are tender and browned in spots and shallots are tender, 4-5 minutes. Remove skillet from heat; stir in brandy (caution: Mixture may ignite). Return skillet to grill. Add broth, Dijon mustard, and thyme leaves; stir to blend.
4. Add cream; simmer until mushroom sauce is slightly thickened, 4-5 minutes. Season sauce to taste with salt, pepper, and cayenne.
5. Spoon sauce onto plates; top each with a filet. Garnish with chives.