Email the mutual contact with a short blurb where you include:
1) A polite note saying that you hope they're doing well. 2) 1-2 sentences about who you want an intro to and why (it should be more specific than "pick their brain") 3) Any urgency or timeframe, and how you hope to speak with them (phone? coffee? etc)
Make sure you write the email in a way that the receiver can literally press "forward" and send it to the end-user directly. This saves the person time whom you're asking a favor from.
I think in general people are usually pretty receptive to making an intro for you. Ask politely, be specific about what you're asking for (contact name or an actual intro or to be mentored etc). Attach your resume/cover letter and ask if the person could forward it.
Can you provide support as to my expertise If asked by a possible Hiring Manager or Recruiter?
Well it depends how much does that person knows/trust me in order for them to introduce me one of their contacts.
The easiest way a person would be willing to introduce me to their contacts are:
a.) If I gave a product/service that he/she was satisfied with
b.) If I have a service/product that is in demand from his/her contacts
From there I wouldn’t have to do too much convincing.
Otherwise I would have to flip the script and present an issue that I need that their contacts can fulfill.
If the person does not know me too well its best that I come up with an interesting conversation to make them feel comfortable, then I would make my move.
A quick introduction would go:
Hello, (the person)
Remember me the (re-explain who you are) from (last interaction)?
I enjoyed (whatever you liked during the last interaction)
(Talk about one of their contacts and why you want to be introduced to them)
Thank you for taking the time to (listening/reading) to me!
(if email attach resume and cover letter) (if on the phone give them your email or phone number)