Place the bell peppers over the flame of a gas burner to char, turning, 3 to 4 minutes per side (or char the peppers under the broiler, turning, 5 minutes). Place in a bowl, cover with plastic wrap and allow the peppers to steam about 10 minutes. Once cool to the touch, remove the peppers from the bowl and pull off the charred skin and stems and remove the seeds. Slice and set aside until ready to use.
Set a large Dutch oven over medium-high heat. Add the olive oil, then the shallots and garlic; season with salt and pepper and cook until the garlic is fragrant and the shallots are translucent, 3 to 4 minutes. Add the red pepper flakes and wine and cook until most of the wine has evaporated, 3 to 5 minutes. Add both cans of tomatoes and the roasted red peppers. Mix well and bring to a steady boil, then reduce the heat so the sauce simmers.
Meanwhile, prepare the kale by removing and discarding the stems; tear the leaves into bite-size pieces. Fold the kale into the tomato sauce along with the chicken and toss to combine. Cover and cook until the kale has wilted, about 5 minutes. If the sauce seems dry, add the chicken stock 1/2 cup at a time. Continue cooking, covered, until the kale is tender, all the flavors have come together and the sauce is thick, 15 to 20 minutes. Taste and season with salt and pepper.
Meanwhile, bring a large pot of salted water to a boil. Add the pasta and cook according to the package directions, 8 to 10 minutes. Reserve about 1 cup cooking water, then drain the pasta and add it directly to the Dutch oven with the finished sauce. Drizzle in some olive oil. Cook over low heat, tossing to coat everything, 2 to 3 minutes. (This allows the last cooking of the pasta to be done in the sauce.) Add a little of the reserved pasta water if needed. Serve the pasta in large bowls and top with spoonfuls of parmesan, more red pepper flakes and another drizzle of olive oil.