Mapping Center Answer:

If you are trying to create gores for making a globe it's easier. If you want to make a gored projection (all gores on the same map), it's more difficult. For the former case, we came up with polyconic projection that you can use for about 15 deg wide gores. We can't automatically restrict the projection to 15 degrees wide (or possibly a bit wider for edges that can be cut) but you could clip the data to the width you want before projecting it.

For the latter case, the steps are:

1. Clip the input data into the area that you want before projecting them.
2. Figure out the offsets you need (false eastings) to slide the individual gores into position.
   
3. • Example, the central gore (false easting = 0), might have min/max easting values of -5000000/+5000000 so we now know how wide the gore will be.
   • For the next gore to the east, the false easting will be +10000000.
4. Remove any coordinate system information. This can also be accomplished by "clearing" the coordinate system from the data frame. Without the coordinate system information, ArcMap can't project the data on-the-fly.