Prepare the mushroom-Parmesan sauce. Using your fingers, mash together 1 tablespoon of the butter and the flour to form a beurre manie ( a quick French sauce thickener). Set aside. In a big, heavy nonstick skillet, heat the butter and oil. Add the sliced mushrooms. Sauté them until they are golden, 10-12 minutes. No need to hover and stir—just let them be and check now and then.
Scatter the garlic around the mushrooms and sauté for another 30 to 45 seconds. Then pour the vermouth over the mushrooms and simmer until most of the vermouth is gone, about a minute.
Add the mushroom broth, the half and half and 3/4-cup Parmesan, and simmering gently, whisk everything together. Add the beurre manie, in bits, and whisk until the sauce starts to thicken. You don’t want it to be super thick, just not thin and runny. Season with salt and pepper, and the sauce is ready.
Assemble the casserole. Preheat oven to 350ºF. Oil a 2-quart deep baking dish. Put a third of the cooked kasha in the dish and smooth it out. Then dot a third of the mushroom-Parmesan sauce all over, spreading it as evenly as you can. Scatter half the fresh spinach on top, press it down with the flat of your hand, then scatter 1/2 cup grated sharp cheddar on top.
Add another layer of kasha, sauce, the rest of the spinach, and a third of the cheddar. Then finish with the last third of the kasha, the rest of the sauce, and the rest of the cheddar. Top with a little more grated Parmesan.
Put the casserole in the oven and bake for 30 to 40 minutes. At the end, if need be, run it under the broiler for a minute or two to brown up the cheese top. Done.
This serves 4 people as a meal, with a salad, or is a side for six+ people. Yes, you can multiply it.