I searched almost a week for this issue and I couldn't find local servise exactly like Nagomi visit for Turkey. I think there isn't such a local service. Both JonathanReez and fkraiem's answers are quite close what you said by the way. Also, I found a web site called https://www.voulezvousdiner.com.
From their FAQ:
What are the benefits of booking through VoulezVousDîner.com?
When you book a dinner at www.voulezvousdiner.com, you are sure to be
greeted by hosts selected for their sense of hospitality, around a
quality meal, prepared exclusively with fresh products (see our
Quality Charter). With VoulezVousDiner, you can choose your dinner and
hosts based on a number of criteria. It’s a way to share cultural
perspectives, anecdotes and common interests (hobbies, sports, travel,
professional activities, etc.) around a menu. It’s an opportunity to
discover a new way of life, immerse yourself in a local culture and
establish new friendly contacts.
You can search for Istanbul with https://www.voulezvousdiner.com/en/search?lat=41.0082376&lng=28.97835889999999&location=Istanbul%2C+%C4%B0stanbul%2C+Turkey&datemin= for example.
After you get their address and phone numbers, you can ask them to prepare together if it is possible. They probably won't say no.
Foods looks delicious by the way.