Your question is about the specific days and specific protocols to be observed for the act of picking the Tulsi leaves off the plant?
Yes , there are these things and is described in a book called Nitya Karma Puja Prakash , which is in hindi. The book in its Deva Puja sectio is describing the rules ,mantras and procedure to pluck plant leaves like Tulsi ,Bilva and other flowers . I am translating the content .
Tulsi , Bilva and flowers should be plucked after bathing. Before plucking one should perform Achamana (taking sip of water) and facing the east direction with folding of hands in Namaskar mudra should recite this mantra.
मा नु शोकं कुरुष्व त्वं स्थानत्यागं च मा कुरु |
प्रार्थयामि वनस्पते ||
O plant i am giving you the trouble by removing you from your place
please don't grief . I am plucking you for the purpose of worship of
gods , I am requesting you to grant me the permission.
Here are mantras to be uttered while plucking first three flowers.
- While plucking first flower - ॐ वरुणाय नम: |
- While plucking second flower - ॐ व्योमाय नम: |
- While plucking Third flower - ॐ पृथिव्यै नम: |
Procedure of plucking Tulsi leaves
Instead of plucking a single leaf of Tulsi Plant , we should pluck the stem with leaves . The inflorescence (Manjiri मंजिरी) is highest among all the flowers . While plucking this inflorescence some leaves should remain with it. Without Shaking the Tulsi plant pluck the inflorescence . By this way the fruit of Puja or worship increases by many a times.
The mantra to be chanted while plucking Tulsi leaves.
तुलस्यमृतजन्मासि सदा त्वं केशवप्रिया |
चिनोमि केशवस्यार्थे वरदा
भव शोभने ||
त्वदङ्गसंभवै: पत्रे : पूजयामि यथा हरिम् |
पवित्राङ्गी कलौमलविनाशिनि || (अन्हिकसुत्रावली)
O Tulsi you are immortal and always dear to Keshava (Vishnu) , I am
plucking you for the Keshava , you will be shine very well. It is
because of you the puja of Hari is possible. Please allow me to pluck
you o sanctified one ,who is destroyer of the bad of Kali age.
Days forbidden for plucking Tulsi leaves.
- In Vaidhriti and Vyatipata Yogas ( time of particular planets).
- On Tuesdays , Fridays and Sundays.
- On Sankranti , jananaShouchya ,MaranShoucya ( Impure time during birth and deaths).
- On Amavasya (New moon) and during two sandhyas (twilights) .
But according to author the puja isn't complete without Tulsi leaves , so in forbidden times one should pick up the leaves which are themselves are dropped on the ground , and Tulsi leaves should not be plucked with wearing sandals and without bathing.
The sources of above can be found in the footnotes of Page No. 109