In a large, deep skillet, heat the olive oil. Add the sliced onions, cover and cook over moderately low heat, stirring occasionally, until they are very soft, about 10 minutes.
Add the minced garlic and cook, stirring, until fragrant, about 3 minutes. Add the Red Cabbage & Beetroot, cover and cook, stirring occasionally, until the cabbage is tender, about 5 minutes. Cut ½ the block of feta cheese into small cubes and add to the skillet to melt.
Meanwhile, in a large pot of boiling salted water, cook the linguine until it is al dente. Drain the pasta. Pour the pasta over the Red Cabbage and season with salt & pepper.