Boil the egg noodles in generously salted water until partially tender, typically 4 to 6 minutes, or according to package directions. Drain thoroughly and set aside.

Preheat the oven to 350°F. Meanwhile, place a large skillet over medium-high to high heat.

Add the chopped onion and ground meat to the hot pan. Season with salt. Sauté, stirring and breaking up the meat, until thoroughly browned—approximately 10 minutes. Ensure the fond (browned bits) does not burn. Transfer the cooked meat and onions to a large mixing bowl, incorporating half of the noodles.

Turn off the heat and immediately add the sauerkraut, including its brine, to the still-hot pan. Use a wooden spoon to deglaze the skillet, lifting the browned remnants. Transfer all contents to the bowl with the meat and noodles.

Introduce the dry mustard, caraway seeds, ground black pepper, and brown sugar into the mixture. Manually crush the whole tomatoes over the bowl, discarding any tough core and reserving the can’s juice for another dish. Thoroughly combine all components.

Firmly pack the prepared mixture into a 9×13-inch casserole dish. Distribute the remaining noodles over the top—using discretion, as not all may be necessary. Evenly layer the grated cheese over the noodles.

Bake uncovered until the cheese is golden at the edges, typically 30 minutes. Let rest for 5 minutes before slicing and serving.