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Cook the rice risotto style until slightly under al dente, season with salt and pepper. Spread out on a sheet pan and let cool.
Preheat the oven to 425°F.
Heat 2 tablespoons of oil in a large high sided pan, add the rice and cook, stirring constantly, until translucent, add the wine and cook, stirring constantly until evaporated. Add the stock, 1 cup at time and cook until slightly under al dente. Spread on a baking sheet and let cool. (can be made a day ahead of time and stored in the refrigerator.
Heat 2 more tablespoons of the oil in a large CAST IRON skillet over medium-high heat until it begins to shimmer. Add mushrooms and cook, stirring occasionally, until mushrooms begin to release their liquid, about 5 minutes. Continue to cook until liquid has completely evaporated and mushrooms are golden brown and tender, 6 to 8 minutes longer; season with salt and pepper and transfer to a plate.
Add the remaining 2 tablespoons oil to pan and add onions, red and poblano peppers and cook, stirring often, until soft, about 3 minutes. Add the garlic and cook, stirring, until fragrant, about 30 seconds; season with salt and pepper. Add the mushrooms back to the pan.
Stir in rice and cook and press down to make an even layer on the bottom of the pan. Stir 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 enough mushroom stock to just cover rice; season with salt and pepper. Cook, without stirring, adding more stock if mixture becomes too dry and rice is not yet cooked. Shake pan after each addition of stock, to help it settle in among the rice and mushrooms. Cook until rice is al dente, 15-20 minutes.
When rice is tender and liquid has been absorbed, continue to cook paella, moving pan around occasionally, until a crust (soccorat) has formed around the sides and bottom of the pan (it will start to smell like toasted rice 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 beginning to wilt. Add 1/4 cup water to skillet and cook, tossing, until completely wilted, about 5 minutes. Spoon over paella.
Make 4 shallow wells in the top of paella and crack 1 egg into each well. Transfer pan to oven and cook until egg whites are just set, about 8-10 minutes. Top with parsley, remaining thyme, and drizzle with chile oil. Serve straight from the pan.