In a large skillet, heat the olive oil, two turns of the pan, over medium-high. Add the mushrooms and cook, stirring often, until browned, about 8 minutes. Add the shallots, season with salt and pepper and stir until the shallots soften, about 2 minutes. Add the stock, stirring to scrape up any browned bits on the bottom of the skillet. Add the kale; stir until wilted, about 2 minutes. Cook, stirring often, until the stock is almost evaporated, about 5 minutes. Remove from the heat.
Meanwhile, preheat the broiler and heat a griddle pan or a large skillet over medium-high. Add a sprinkle of water to the pan. Working in two batches, toast the naan until blistered in spots, about 2 minutes per side.
Arrange the naan on a large, rimmed baking sheet; top with the mushroom mixture and cheese. Broil until the cheese is bubbly and browned, 2 to 3 minutes.