I think your confusion stems from the fact that you are using the term "employment" whereas you should be using the term "engagement." Further, I am guessing that in order to maximize the post-tax money into the hands of the person, he/she has been engaged under 194(c) and has been shown as a Contractor, even though possibly he/she is an employee in all other respects.
Assuming that my guesses are correct, you need to bear in mind that the person you have engaged is a Contractor and hence cannot be an employee. If a person is not an employee, there can be no case for seeking any salary slip/pay slip.
If I were to draw a parallel for purposes of an example, let us assume that you have engaged a person as a media planner to handle your advertisements and media coverage. It is likely that this contract would be open-ended, i.e., in perpetuity till revoked. Also, this person would be paid a certain amount on a monthly basis for services rendered. However, this person cannot claim or ask for a salary slip/pay slip. The person is either an employee (in which case 194(c) would not apply and TDS at normal rates for an employee would apply) or the person is a contractor (in which case 194(c) would apply). You cannot have a situation where a person is a contractor and at the same time seek a monthly pay slip like an employee.
Regards,
Raju Bhatnagar