If knower and known are both the One pure essence,
What are the aspirations in this handful of dust?

What are the aspirations? – What is the nature and purpose of the aspirations?

We could rephrase the question thus:

If knower and Known are one essence, why does this handful of dust (human self) so desperately yearn to know?

The question is really about the nature and purpose of ‘ishq[1] , which is central to Shabistari’s answer. The reason this desperate yearning exists is because from the Known you gained your substance, life and faith; at creation you acknowledged Him Lord and swore fealty (Q7:172); and because the Known gave you the faculty of ‘ishq so that you would passionately desire to know Him (couplet 429, Lahiji’s explanation).

The purpose of  ‘ishq is told by Abu Huraira, who narrated a hadith in which the Prophet said,

…And the most beloved things with which My slave comes nearer to Me, is what I have enjoined (frD) upon him; and My slave keeps on coming closer to Me through performing nawãfil (praying or doing extra deeds besides what is obligatory) till I love him,

so I become his sense of hearing with which he hears, and his sense of sight with which he sees …

— Bukhari, vol 8, hadith 509[2]

Thus, joining the dots, you yearn to know because as a result of this desperate yearning placed in you at your creation, the Known knows Himself through you as he becomes your seeing, your knowing, your doing, if you allow it. In that moment both knower and Known are seen (couplet 431). So look at yourself, you who are a reflection of Him. Look at yourself and ask of yourself all that you desire to know (couplet 432–433).

[1] See Dara’s explanation of ‘ishq here, http://www.untiredwithloving.org/prehension.html#ishq_self_entanglement

[2] As cited by Supererogatory Works, nawāfil, http://www.livingislam.org/nawafil.html