Rinse the green beans, snap off the stem end, and then break each bean into one to two inch sections. Place the green beans in a medium pot and add about one inch of water. Cover the pot and bring it up to a boil over high heat. Allow the beans to cook in the steam until they are tender-crisp (about five minutes), or until they are to your desired tenderness. Drain the beans in a colander and set them them aside.
Preheat the oven to 375 degrees. Dice the onion and cook it in a large skillet with 2 tablespoons of butter over medium heat until the they are soft and translucent. While the onions are cooking, clean and slice the mushrooms.
Add the sliced mushrooms to the skillet with the onions. Season the mushrooms with 1/8 tsp of salt and 1/4 tsp thyme. Continue to cook and stir the mushrooms until they have softened and most of the moisture released from them has evaporated from the bottom of the skillet.
Add the flour to the skillet and continue to cook and stir for 2 minutes. The flour will form a paste as you stir and cook. After 2 minutes, whisk in the vegetable broth until the flour has fully dissolved in.
Allow the mixture to come up to a simmer, at which point it will thicken slightly. Whisk in 1 tablespoon of grated parmesan.Add the cooked and drained green beans to the skillet with the mushroom gravy. Stir to combine the mixture.
In a separate bowl, melt the remaining 2 tablespoons of butter (about 30 seconds in the microwave on high). Stir in the panko bread crumbs, then stir in the remaining tablespoon of grated parmesan. Stir will until everything is evenly coated in butter.
Sprinkle the parmesan panko topping over the green beans in the skillet. Transfer to the oven and cook for 30 minutes, or until the panko topping is golden brown. Serve hot.
*I use Better Than Bouillon soup base to make vegetable broth rather than using canned or boxed broths. *If you do not have an oven safe skillet, transfer the green beans and mushroom gravy to a casserole dish (8x8 inch), top with the panko breading, then bake as directed.