Is the process of rebuilding the Brisk Menu, to add a category such as "Internet" or "Graphic", a task which is considered part of the underlying Ubuntu OS, or is it considered part of the Display Manager.
I ask because, after installing first MATE Desktop (on top of Ubuntu Server), then Firefox or Gimp on the Ubuntu server, there is no category listed for either "Internet" or "Graphics" !!!
Not sure if this will help, but you could try:
sudo update-desktop-database
Thank you for responding, Dave, but that utility (along with other tools in the "desktop-file-utils" package) are addressing an issue at a much lower level than what I am trying to deal with.
My issue is how does the OS handle the installation of a new application (Firefox, GIMP) such that once installed, along with any well-formed "*.desktop" file,
how does it create the Brisk Menu category heading "Internet" (if missing, which in my case it is), and
how and where does it place that "*.desktop" file such that it is "plugged-in" to that Brisk Menu structure
for subsequent visibility when you press on the Roundel Icon on the Panel, and offer the newly added selection (Firefox) in the expected "pop-up" window-pane that lists all the "launchers" for the various installed Applications (in this case, specifically, Firefox)?
Under installed files for
there is the following file listed:
That "BriskMenuFactory" is a strong hint of something which I can't seem to locate when searching on "Description and Name" within Synaptic. 
After you’ve installed the app, is their any reason you can’t use the “Menu Editor” tool Main Menu in the Control Center to create any missing categories and make the menu item? That would seem to be the highest level.
Thank you, Walter (?). I appreciate the suggestion. I was aware of that tool, and that I could do all the required edits and "customization" manually.
However, I don't want to do that, because that would imply requiring Users to replicate my attempts to "automate" tasks for a custom build (Ubuntu-Server + MATE Desktop + reduced User-selected set of Apps), then I will have to abandon the attempt to come up with at UbuntuMATE-like alternate that could step in if UbuntuMATE were to become "frozen", as a flavour, for lack of development.
Under UbuntuMATE, the process would somehow automatically add a missing category, then deposit the App-specific launcher in the required menu "slot".
I'm trying to find the specific mechanism I need to install as the first Application after the server OS base followed by the MATE Desktop install.
Being easily prone to intriguing rabbit holes, I did a little research. (Disclaimer: I'm an amateur, so verify this info for yourself before you do something irreparable.)
It seems the various categories that appear in the menus differ slightly for each menu. For example, my Brisk menu has a top-level Administration category, but the Compact menu provides the Administration category as a cascaded second-level category under the System category.
Apparently, the structure (category-wise) of the various MATE menus is controlled by the ".menu" files in /etc/xdg/menus/. As I understand the process, these ".menu" definition files are created/installed by the mate-menus package. Can you check that this package has been installed?
Even when those ".menu" files exist, a particular application's ".desktop" file can override the menu category in which it will appear, or even prevent it from being shown at all. Do you know whether you get the same ".desktop" file regardless of whether Firefox is being installed on Ubuntu server or Ubuntu desktop, or is there a conditional branch in the installation process that installs one Firefox ".desktop" if the target is a server and another if the target is a desktop?
Evidently, installing MATE desktop on a non-desktop Ubuntu installation is more complex than I realized. It might be more reliable to start with a non-MATE Ubuntu desktop installation and then switch the DE to MATE, but that may not be compatible with your goals.