You can add a free stopover (in the civil aviation sense, a stop >24 hours enroute to a final destination) on any EK ticket (excl. hotel & visa, but to stop for 3 days on the way has no additional fare vs to connect direct).

Dubai Connect becomes available when there is no shorter connection on the same day for any combination of flights (incl. a earlier/later departing flight from the origin), even if the shorter connection is more expensive. Given EK's schedules it is unfortunately kind of difficult to find itineraries with such long layovers (used to be easier) as EK often has multiple flights a day. If you're flying from CPT take into consideration that the 3 daily flights ex CPT arrive between midnight and 8am in DXB, so your best bet for Dubai Connect is to a destination where the only flight departed between 4pm and 11pm DXB time, thus forcing a long layover.

The other option is to book relatively last minute when all other flights are sold out, this has worked for me ex CPT before, but prices overall will be higher and of course harder to plan ahead of time (and a gamble if you're doing it just to get DC).