Pat steak dry with paper towels and season with salt and pepper. Heat 1 tablespoon oil in large stainless steel, cast iron, or aluminum skillet over medium-high heat until very hot but not smoking. Add steak tips in a single layer and sear on all sides until the exterior is nicely caramelized, 6 to 8 minutes. Depending on the size of your skillet, you might need to do this in 2 separate batches. Transfer the steak to a serving dish in a single layer, and loosely cover with foil.

Adjust the heat to low, then add the shallots to the pan along with a pinch of salt (if the bottom of the pan isn't shiny, add a small splash of oil first). Stir, using a spatula to scrape up the brown bits from the bottom of the pan. If the brown bits won’t detach or look like they might burn, deglaze the pan with 1 tablespoon of water to help loosen them.

After 1-2 minutes, add the mushrooms and garlic, turning the heat up to medium-low, and cook until the mushrooms are soft, 3-4 minutes.

Stir in the stock, cream, and any juices that have accumulated around the steak. Scrape up any additional brown bits if they appear. Simmer the sauce until slightly thickened, about 5 minutes.

Remove from the heat and whisk in 1/2 cup of the blue cheese. Pour the sauce over the steak, and top with the remaining cheese and chives. Serve immediately.