Heat 2 tablespoons of the olive oil in a 12-inch paella pan or ovenproof skillet over medium-high heat. Cook the mushrooms, tossing occasionally, until they release their liquid and are slightly brown, 4 to 6 minutes. Add the onion, bell pepper, poblano and garlic, and cook, stirring constantly, until they soften, about 3 minutes; season with salt and pepper.
Add the rice and cook, stirring constantly, until the rice is translucent, about 3 minutes. Add the wine and cook, stirring, until completely evaporated, about 2 minutes. Return the mushrooms to the pan, stir in the 1 teaspoon thyme, and add enough mushroom stock just to cover the rice; season with salt and pepper. Cook, without stirring, adding more stock as needed to keep the rice moist while cooking, until the rice is al dente and all the liquid is absorbed, 15 to 20 minutes. (Shake the pan after each addition of stock to distribute it evenly.)
Continue to cook the paella, occasionally moving the pan around the burner to make sure all areas get evenly heated, until a crust (socarrat) forms around the sides and bottom of the pan (the rice will smell toasted and make a light crackling sound), 6 to 8 minutes.
Make 4 shallow divots on the surface of the paella and crack an egg into each divot. Transfer the pan to the oven and bake until the egg whites are just set, 8 to 10 minutes.
Meanwhile, heat the remaining 2 tablespoons of olive oil in a large skillet over medium-high heat. Add the kale, season with salt and pepper, and cook, tossing, until slightly wilted. Add 1/4 cup water to the skillet and cook, tossing, until the kale is completely wilted, about 5 minutes longer.
Arrange the kale around the eggs, drizzle with the Calabrian chile oil and sprinkle with the parsley. Serve the paella in the pan.
Place porcini mushrooms in a medium heatproof bowl and cover with the boiling water; let sit 30 minutes. Line a fine-mesh sieve with 2 layers of paper towel and set over another medium heatproof bowl; pour mushroom mixture through, letting the towels catch any grit from the mushrooms. Set aside the porcini mushrooms and their soaking liquid.
Heat the oil in a large saucepan over medium heat. Cook the onion, carrot, and celery, stirring occasionally, until softened, 5 to 7 minutes. Add the cremini mushrooms, parsley, thyme, peppercorns, salt, reserved porcini mushrooms and their soaking liquid, and 6 cups cold water. Bring to a boil, reduce the heat, and simmer for 30 minutes.
Strain the stock into a clean saucepan and keep warm over low heat (you should have about 6 cups). Stock can be made 4 days ahead. Cover and refrigerate. Reheat before using.
Calabrian Chile Oil:
Combine the chiles, oil and thyme in a blender and blend until smooth; season with salt and pepper. Can be made a few days in advance, tightly covered and refrigerated. Bring to room temperature before using.