Preheat the oven to 350 degrees F (175 degrees C).
Bring a large pot of lightly salted water to a boil. Add penne and cook, stirring occasionally, until tender yet firm to the bite, about 11 minutes. Drain and return to the pot.
Remove beef roast from the package and place in a bowl, reserving juices. Shred into bite-size pieces using 2 forks.
Melt butter in a large skillet over medium heat. Add mushrooms, onion, and garlic; cook and stir until tender, 4 to 5 minutes. Stir in flour and tomato paste until combined. Stir in reserved meat juices, broth, Worcestershire sauce, paprika, salt, and pepper. Cook and stir until thickened and bubbly, 6 to 8 minutes. Remove from heat and stir in 1/2 cup sour cream.
Stir shredded beef and mushroom mixture into the pot of cooked pasta. Spoon mixture into a 9x13-inch baking dish. Cover with aluminum foil.
Bake in the preheated oven until heated through, about 30 minutes.
Combine remaining sour cream and horseradish in a small bowl. Add dill. Spoon on top of the casserole.