Like Ven. Pesala, I’ve never heard of any tradition of memorizing and reciting this text. The only monks I can imagine doing so are those (exceedingly few) who aspire to be vinayadharas. Still, if this is what you want to do, perhaps you could make a start by memorizing those portions of the Mahākhandhaka which are in more or less regular liturgical use and which most senior monks are likely to have memorized. It shouldn’t be hard to find recordings of them at Youtube. They are:
1. Paṭiccasamuppāda anuloma: “avijjāpaccayā saṅkhārā, saṅkhārapaccayā viññāṇaṃ…”
2. Paṭiccasamuppāda paṭiloma: “avijjāyatveva asesavirāganirodhā saṅkhāranirodho…”
3. Three Udānas: “yadā have pātubhavanti dhammā…”
4. Dhammacakkappavattana Sutta.
5. Anattalakkhaṇa Sutta.
6. Ādittapariyāya Sutta.
8. Tisaraṇagamana: “buddhaṃ saraṇaṃ gacchāmi…”
9. Assaji’s gāthā to Sāriputta: “ye dhammā hetuppabhavā…”
10. Upasampadā formulas: “suṇātu me, bhante, saṅgho, ayaṃ itthannāmo itthannāmassa āyasmato upasampadāpekkho…”
11. Cattāro nissayā: “piṇḍiyālopabhojanaṃ nissāya pabbajjā…”
12. Cattāri akaraṇīyāni: “upasampannena bhikkhunā methuno dhammo na paṭisevitabbo…”
Khandhānaṃ rāsaṭṭhaṃ, āyatanānaṃ āyatanaṭṭhaṃ,
Dhātūnaṃ suññaṭṭhaṃ, indriyānaṃ adhipatiyaṭṭhaṃ,
Saccānaṃ tathaṭṭhaṃ aviditaṃ karotītipi ‘avijjā’.
It prevents knowing the meaning of heap in the aggregates, the meaning of actuating in the sense-bases, the meaning of voidness in the elements, the meaning of predominance in the faculties, and the meaning of suchness in the truths, thus it is called ‘ignorance’.
(Visuddhimagga XVII. 43)