Heat the oven to 400°F and position 2 racks toward the center with a few inches in between them. Line 2 rimmed baking sheets with foil and smear each with 1 tablespoon oil. Remove the stems from the mushrooms and save them for another use (like stock). Slice the caps crosswise as thin as you can manage.
Spread the mushroom slices on the prepared pans in a single layer (it’s okay if they overlap a bit) and sprinkle with the paprika, salt, and pepper. Transfer the pans to the oven and bake, switching racks and rotating the pans halfway through, until the mushrooms release their water and the pan is almost dry again, 20 to 30 minutes. Lower the heat to 325°F and continue to cook the slices, rotating again to ensure even browning, until they dry and shrivel a bit, and release easily from the pan, another 5 to 10 minutes. Transfer to a wire rack to cool completely.
Meanwhile, use the slicing blade on a food processor or a knife to thinly slice the Brussels sprouts.
Put 2 tablespoons oil in a large skillet over medium-high heat. When it’s hot, add the onion and garlic. Cook, stirring often, until the onion softens, 3 to 5 minutes. Add the rice, lower the heat to medium, and stir until the rice is glossy, completely coated with oil, and starting to toast, 3 to 5 minutes. Sprinkle generously with salt and pepper, then add the apple cider. Cook, scraping any browned bits from the bottom of the pan, until the cider has mostly evaporated, just a minute or 2. Turn the heat down to low and add the stock and Brussels sprouts. Bring to boil, lower the heat to a steady, gentle bubble, stir once or twice, then cover.
From this point, the rice will take about 40 minutes to become tender; check after 30 minutes to make sure there’s enough liquid and, if not, add about 1/4 cup more. When the rice is tender, turn the heat to the absolute minimum (if you have an electric stove, turn the heat off and let the pan sit on the burner), and let rest, covered, for another 15 to 30 minutes. Add the remaining 2 tablespoons oil and the dill and fluff with a fork. Taste and adjust the seasoning, fluff again, garnish with the mushroom bits, and serve.