Preheat the oven to 375 degrees . In a large pot of boiling, salted water, cook the egg noodles according to package directions until al dente. Drain and return to the pot. Stir in the kielbasa and 1 stick butter.
Meanwhile, in a large bowl, stir together the arugula, eggs, cottage cheese and sour cream. Add to the noodle mixture, season with salt and pepper and toss. Spread evenly in a greased 9-by-13-inch baking dish. Bake until the crust is golden and the casserole is set, 30 to 35 minutes. Let stand for 10 minutes before slicing.
Meanwhile, in a medium skillet, melt the remaining 1 tablespoon butter over medium-high heat. Add the radishes and vinegar; season with salt and pepper. Cook until crisp-tender, 5 to 7 minutes. Serve with the noodle bake.