My tired brain has a couple of ideas. Whether or not they seem stupid in the morning is another matter.
1. Plate segments are individual, unbonded "cousin" spren that are attracted by the Radiant's spren/abilities/personality, that are somehow turned into the pieces. Regrowing is just letting the spren congregate to their friends associated with intact pieces, to fill in what's missing.
-or-
2. Cultivation + [something]. (though what entertains this idea most is that I don't know anything about that Shard, and I like the mystery XD )
Also, what's saying regular Plate won't glow when a full-fledged Radiant wears it? The only times it could have glowed was when Kaladin had the helmet on his hand, but that wasn't a full set, and he wasn't worried about the glow since the sun was bright, anyway. Maybe a full set would show the excess breathed Stormlight like in Dalinar's visions?