Do you mind my...going?

Sí, es una estructura sorprendente, especialmente si la traducimos al castellano (o al francés.) Aquí ese gerundio es "nombre, "sustantivo":
"I dislike your complaining all the time."
"We enjoyed her singing."
"The teacher minds my smoking in the classroom."
"Your being late all the time is not acceptable."
"My meeting her on the street was a coincidence."
I would appreciate your speaking to him about his credit."
Es una estructura fantástica una vez nos hacemos a ella. Estos ingleses tienen unos descubrimientos gramaticales que nos dejan boquiabiertos.