Preheat the oven to 425 degrees. Place the broccolini on a large rimmed baking sheet; season with salt and pepper. Toss with the garlic, chile and a fat drizzle of EVOO. Spread out in an even layer and roast until the stalks are tender and the edges are crispy, 15 to 20 minutes.
In a small saucepan, bring 1 1/2 cups chicken stock and the porcinis to a boil over medium-high heat. Reduce the heat to low. Let simmer to reconstitute the mushrooms while you make the chicken.
Season the chicken. In a large, deep skillet, heat the olive oil, two turns of the pan, over medium-high. Add the chicken and cook until browned, turning once, 7 to 8 minutes; transfer to a plate. Add the crimini mushrooms to the skillet and cook, stirring often, until well browned, about 5 minutes. Add the shallots and thyme; season. Cook, stirring often, until the shallots soften, 2 to 3 minutes. Sprinkle with flour and stir to coat the mushrooms. Add the wine and reduce the heat a bit. Using a slotted spoon, remove the porcinis from the stock and chop. Add the porcinis to the skillet. Pour about 1 cup porcini stock into the skillet, leaving any sediment behind. Bring the sauce to a simmer. Slide the chicken and any juices back into the skillet and simmer at low bubble, turning the chicken occasionally, until the chicken is cooked through, 8 to 10 minutes more. Stir in the truffle oil and parsley.
Meanwhile, in a medium saucepan, bring the remaining 3 cups chicken stock and the milk to a low boil over medium-high heat. Whisk in the polenta, then keep whisking until the polenta is thick and tender, 2 to 3 minutes; season. Stir in the cheese, butter and honey.
Spoon a bed of polenta into shallow bowls and top with the chicken and mushrooms. Serve with the broccolini.