I don't think that these composite plots are built-in to JMP, but you can create something similar by combining reports. It might be helpful to create an ID column to designate which rows should be combined together. For the effect of pie-in-pie, this could mean copying your category column and changing some of them to "Other", or it could be more general if there are other meaningful groups.
Once you create two graphs (one using the ID column, one using the full category columns), you can use Combine Windows to merge them together, and choose the ID column as a Selection Filter. Below is what I get using age as the grouping column with the Big Class sample data. With 16-year-olds selected on the left, the individual 16's are shown on the right.
One advantage of using selection filters is that the behavior is dynamic - you can select any category on the left and the plot on the right will update. You can also select multiple categories, so if you want to see 12- and 13-year-olds in one plot, you can select them both. You are also not limited to Pies and Bars - you can use any JMP reports that work for your data.
There are more selection filter examples in Using Selection Filters in JMP 12 . This article uses Application Builder to create the report, but starting in JMP 13 the functionality is built directly into Combine Windows.