Put a large, heavy-bottomed skillet over high heat. While the pan is heating, season the steaks with salt and pepper on both sides.
Add the oil, making sure there is enough to coat the bottom of the pan. Add the steaks and sear on 1 side for 2 minutes. Turn them over and sear on the other side for 3 minutes. (This will cook the steaks to medium-rare.)
Lower the heat and add 3 tablespoons of the butter, the garlic, and thyme. Baste steaks with the melted butter on both sides for 1 more minute. Remove the steaks from the pan and let them rest for 2 minutes.
While the steaks are resting, remove the garlic and thyme from the pan and wipe out the excess butter. Add the remaining tablespoon of butter and let it brown over high heat. Add the shallots and cook for a few seconds. Add the spinach and saute, stirring vigorously, until it is wilted, about 1 minute.
Put a bed of spinach onto each of 4 plates, place a steak on top, and serve.