Coat the bottom of a large cast iron skillet with olive oil. Heat the skillet on high heat until the pan is very hot.
Salt and pepper the steaks on both sides.
Add the steaks to the pan and sear each side for about 2 minutes to form a crust. Make sure the steaks are not touching, or they will steam instead of searing well.
Add a tablespoon of butter and a sprig of thyme to the top of each steak, leaving them in the skillet, and transfer them to the oven to cook for another 4 minutes.
Meanwhile, heat a small saucepan over medium heat and add the persimmons. Add the lemon juice and honey. Cook and stir for 2 to 3 minutes, just until the fruit begins to break down. Add the lemon zest and remove from heat.
Remove the steaks from the oven. Transfer to a cutting board and let rest. Discard the thyme sprigs as they will be charred but their flavor has infused the meat.
Place the same skillet over medium heat on the stove. Deglaze the pan with the beef broth and the wine.
Add the shallot and cook for about 2 minutes until translucent but not browned.
Add the chard and toss gently with tongs. Cook for only a minute or two, just until it begins to wilt.
Place a mound of chard on each of two plates. Set the steak so that it is leaning against the chard. Spoon the compote over the steaks and serve.