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Place porcini mushrooms in a medium bowl and cover with 1 cup boiling water; let sit 30 minutes. Line a fine-mesh sieve with 2 layers of paper towel and set over another medium bowl; pour mushroom mixture through (towels will catch any grit from mushrooms). Set aside porcini mushrooms and soaking liquid.
Heat oil in a large saucepan over medium heat. Cook onion, carrot, and celery, stirring occasionally, until softened, 5–7 minutes. Add crimini mushrooms, parsley, thyme, peppercorns, salt, reserved porcini mushrooms and their soaking liquid, and 6 cups cold water. Bring to a boil, reduce heat, and simmer 30 minutes.
Strain stock into a clean saucepan and keep warm over low heat (you should have about 6 cups).
DO AHEAD: Stock can be made 4 days ahead. Cover and chill. Reheat before using.
Preheat oven to 425°. Heat 2 Tbsp. oil in a 12” paella pan or skillet over medium-high heat. Cook mushrooms, tossing occasionally, until they release their liquid, about 4 minutes. Continue to cook until liquid evaporates and mushrooms are golden brown and tender, 5–7 minutes longer; season with salt and pepper and transfer to a plate.
Add 2 Tbsp. oil to pan and cook onion, bell pepper, and poblano chile, stirring often, until soft, about 3 minutes. Add garlic and cook, stirring, until fragrant, about 30 seconds; season with salt and pepper.
Add rice and cook, stirring constantly, until rice is translucent, about 3 minutes. Add wine and cook, stirring, until completely evaporated, about 2 minutes. Return mushrooms to pan, stir in 1 tsp. thyme, and add mushroom stock just to cover rice; season with salt and pepper. Cook, without stirring, adding broth as needed to keep rice moist while cooking, until rice is al dente and all liquid is absorbed, 15–20 minutes. (Shake pan after adding more stock to distribute evenly.)
Continue to cook paella, occasionally moving pan around burner to make sure all areas get evenly heated, until a crust (socarrat) forms around the sides and bottom of the pan (rice will smell toasted and make a light crackling sound), 6–8 minutes.
Meanwhile, heat remaining 2 Tbsp. oil in a large skillet over medium-high heat. Add kale, season with salt and pepper, and cook, tossing, until slightly wilted. Add ¼ cup water to skillet and cook kale, tossing, until completely wilted, about 5 minutes longer. Spoon over paella.
Make 4 shallow divots in the top of paella and crack an egg into each divot. Transfer pan to oven and cook paella until egg whites are just set, 8–10 minutes.
Top paella with parsley and more thyme and drizzle with Calabrian Chile Oil. Serve in pan.