Aster Protoflorian uses Exoproofing to become imbued with one element while becoming weak to its opposite. After achieving the first stagger with elemental abilities, it is possible to stagger it a second time. During the first stagger it will be unable to retaliate.
Along with the Beggar's Beads, depending on what elemental form Aster Protoflorian is in can influence the type of abilities dropped.
A possible strategy to approaching the Aster Protoflorian is to rush into battle with the Windwalker Bracelet or similar "En-" accessory, begin with a bout of Imperil and Deprotect and start attacking. Overclocking during this period will likely finish it straight away, making for an easy source of AMP Chips to complete sidequests to obtain the useful Soldier of Peace garb.
Otherwise, it is always a good idea to engage in battle with the creature having all level 2 elemental abilities to maximize the stagger potential after it uses Exoproofing.
The Aster Protoflorian is an automaton created by Cocoon's technology, as mentioned by Lightning in Jagd Village, and is engaged as a boss in Final Fantasy XIII by Lightning and Hope. The model used is the same as the original game, however, it has developed a green moss on its metallic carapace to highlight its advanced age.