How long a stopover do you want? Sometimes you can get up to a 24hr stopover by booking direct but finding the longest connection (i.e. enter JFK-BOM but find the option with the longest MXP connection).

Otherwise an alternative would be to book the flight but in Flex/Flex Plus then later call and try change the MXP-DXB-BOM segments to a later date. The problem I could foresee is that if you don't usually exit the aircraft in MXP (not sure how the transfer works there) that you would theoretically have a single flight from JFK-DXB booked regardless of MXP stop, and to add the stop would require two one-ways.