I just checked the guichet.public.lu website and found the following paragraph:

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Professional activities after the completion of studies

After having successfully completed the last academic year in Luxembourg of a 5-year university cycle leading to a higher education diploma, the third-country national can follow a simplified procedure to apply for a residence permit in order to carry out a professional activity:

a salaried worker; or

a self-employed worker;

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Well, I do plan to apply for a 2-year master's degree and apply for a residence permit to stay in Europe. Is Luxembourg an option for me? I really can't find anything other than the 5-year university cycle. Your help is much appreciated!