Heat the oven to 375 F (190 C). Grease a 9x13-inch baking dish or a 3-quart casserole with a little olive oil or vegan butter. Slice the potatoes about 1/8 inch thick with a mandoline, a food processor, or the slicing side of a box grater, and leave the slices unrinsed so their surface starch stays on.

Heat the olive oil or vegan butter in a large skillet over medium heat. Add the onion with a pinch of salt and cook, stirring now and then, until soft and lightly golden, 5 to 6 minutes. Stir in the garlic and thyme and cook for 30 seconds, until fragrant.

Sprinkle the flour over the onion and stir constantly for 1 minute, until it smells lightly toasted and no dry powder remains. Pour in the oat milk in a slow, steady stream while whisking, then add the vegetable broth the same way, scraping up anything stuck to the pan.

Add the nutritional yeast, Dijon mustard, smoked paprika, 1 teaspoon of salt, and several grinds of black pepper. Bring the sauce to a gentle simmer and cook, whisking often, until it thickens enough to coat the back of a spoon, 5 to 7 minutes. Taste and adjust the salt; the sauce should be a little more seasoned than you want the finished dish to be.

Spread a thin layer of sauce across the bottom of the baking dish. Arrange a third of the potato slices in an even layer, overlapping them slightly, and season with salt and pepper. Spoon over a third of the sauce, then repeat twice more, finishing with the last of the sauce on top. Press the potatoes down with the back of a spoon so the liquid sits just level with the top layer.

Cover the dish tightly with foil and bake for 40 minutes. Remove the foil and bake for another 25 to 30 minutes, until the top is golden and bubbling and a knife slides into the center with no resistance. If the top browns before the potatoes are tender, cover it again and keep baking. Let the dish rest for 10 minutes before serving, then dust with a little more smoked paprika.