Preheat the oven to 450 degrees F. Mix the caraway seeds, paprika and 1/2 teaspoon each salt and pepper in a small bowl. Put the pork in a large ovenproof skillet and rub all over with 1 teaspoon olive oil; season with the spice mixture. Roast 15 minutes, then flip and scatter the mushrooms around the pork. Continue roasting until the pork is browned and a thermometer inserted into the center registers 145 degrees F, about 10 more minutes.
Meanwhile, heat the remaining 1 tablespoon olive oil in a Dutch oven over medium-high heat. Add the garlic and cook, stirring, until it starts browning, about 1 minute. Stir in the shredded vegetables, 1 cup chicken broth and 1/2 teaspoon salt. Cover and cook until the greens are wilted, about 10 minutes. Stir in the dill and lemon zest; keep warm over low heat.
Transfer the pork to a cutting board. Put the skillet with the mushrooms over medium heat; add the remaining 1/2 cup chicken broth and scrape up any browned bits from the bottom of the pan. Bring to a boil, then add the heavy cream. Return to a boil, then immediately remove from the heat. Slice the pork and top with the mushroom sauce. Serve with the vegetables.