Prepare the Pan-Seared Rib-Eye: Preheat oven to 400°F. Heat oil in a large cast-iron skillet over medium-high until shimmering and just beginning to smoke, 2 to 3 minutes.
Pat steak dry with paper towels; sprinkle with salt and pepper. Add steak to skillet; cook, undisturbed, until evenly browned, 4 to 5 minutes.
Reduce heat to medium, and flip steak. Add butter, garlic, and thyme. Cook, basting steak constantly with butter mixture, until garlic caramelizes and sides of steak are no longer pink, 4 to 5 minutes.
Transfer skillet to preheated oven, and cook until a thermometer inserted into thickest portion of steak registers 130°F, 6 to 9 minutes.
Transfer steak to a cutting board, and let stand, uncovered, 5 minutes.
Prepare the Mushroom-Hunter’s Sauce: Melt 2 tablespoons of the butter in a large skillet over medium. Add mushrooms, and cook, stirring occasionally, until lightly browned, 5 to 6 minutes.
Add shallot; cook, stirring constantly, until tender, about 1 minute.
Add wine, and cook, undisturbed, until almost dry, about 2 minutes.
Add tomato puree and stock; cook, stirring occasionally, until mixture bubbles and thickens, 8 to 10 minutes.
Remove skillet from heat. Add remaining 6 tablespoons butter to mushroom mixture, 1 tablespoon at a time, whisking constantly after each addition, until mixture is well combined and sauce is glossy.
Stir in thyme, salt, and pepper; remove from heat.
To Serve: Cut steak into 1/2-inch-thick slices. Spoon Mushroom-Hunter’s Sauce over sliced steak; garnish with thyme leaves. Serve immediately.