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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.
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
Meanwhile, heat 1 Tbsp. 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.
Add cream; simmer until mushroom sauce is slightly thickened, 4–5 minutes. Season sauce to taste with salt, pepper, and cayenne.
Spoon sauce onto plates; top each with a filet. Garnish with chives.
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