Make the stew:
Position a rack in the center of the oven and heat the oven to 350°F. Pour the broth into a small saucepan and add the mushrooms. Bring to a simmer over medium-high heat. Remove from the heat, cover, and steep for at least 15 minutes.
Meanwhile, heat 2 Tbs. of the oil in a heavy stew pot or shallow 5-quart Dutch oven over medium-high heat. Pat the steaks dry, season lightly with kosher salt and pepper, and put only as many in the pan as will fit without crowding. Sear the steaks, flipping once, until nicely browned, 3 to 4 minutes per side. Set aside on a platter, and repeat with the remaining steaks.
Lower the heat to medium, and if the pan looks dry, add the remaining 1 Tbs. oil. Add the carrots, celery, onions, and thyme. Season with salt and pepper and cook, stirring occasionally, until the vegetables begin to soften, about 7 minutes. Stir in the tomato paste, and cook for a few minutes. Add the flour, stirring to blend, and cook for another minute. Add the wine, bring to a simmer, and reduce the heat to low. With a slotted spoon, scoop the mushrooms from the broth and transfer to a cutting board. Coarsely chop the mushrooms and add them to the vegetables. Slowly add the broth, being careful to hold back the last few tablespoons, which may contain grit from the mushrooms.
Cut the steaks into 1/2- to 3/4-inch cubes and add to the pot, along with any juices. Cover tightly and transfer to the oven. Cook, stirring once or twice, until the meat is tender, about 1 hour. Season to taste with salt and pepper. Set the stew in a warm place, and increase the oven temperature to 375°F. (Or if making ahead, let cool, then cover and refrigerate.)
Make the topping:
About 30 minutes before the stew is ready, put the potatoes in a large saucepan and cover by an inch with cold water. Add 1 tsp. kosher salt. Bring to a simmer over medium heat, partially cover, and simmer until the potatoes are easily pierced with a skewer, about 20 minutes. Drain, and return the potatoes to the saucepan. Put the pan over low heat and shake or stir the potatoes until a floury film forms on the bottom of the pot, 1 to 2 minutes.
Using a ricer, food mill, or potato masher, mash the potatoes. Stir in the 5 Tbs. of butter with a broad wooden spoon. Once the butter is thoroughly absorbed, add the milk or cream in three parts, stirring vigorously between additions. Season to taste with salt and pepper.
Assemble and bake:
Lightly butter a shallow 3-quart baking dish. Spoon the stew into the baking dish. Spread the potatoes on top in an even layer—you don’t need to spread them all the way to the edge. Dot the top with the remaining 2 tsp. butter. Bake at 375°F until the stew is bubbling around the sides, and the top is lightly browned, 35 to 45 minutes (45 to 55 minutes if the pie has been refrigerated).
