NOTE:

'However' is also means "whatever way or manner"

*⃣1- Do you mind my making a recommendation?
*⃣2- Do you mind me making a recommendation?

Which pronoun should come before a gerund (objective pronouns or possessives)

Both correct but today a possessive form is usually recommended.
Remember a possessive pronoun in this case is usually the better choice to consider what you're implying.

*when you ask if someone minds "me making a recommendation" you're asking if they object to you.

*when you ask if someone minds "my making a recommendation" you're asking if they object to the act of recommendation.

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