“Combined cap and baseball mitt,” A patent by Richard Villalobos, 1983.
This is meant for fans, not players. The idea is that if a foul ball comes towards you, you may not have time to grab a glove you’ve stashed at your feet. Rather, you quickly slip your hand into the cap-mounted glove and snag the foul with your hat still attached to the back of your hand.
Quomodocumque 
_Rather, you quickly slip your hand into the cap-mounted glove and snag the foul with your hat still attached to the back of your hand._
My understanding was that the crown of the hat is meant to go in front of your hand. For example Fig. 7 shows the protective pad inside the crown, and this is where the ball is supposed to end up.
Either way, it doesn’t strike me as very practical: it seems that the bill of the hat would make it very awkward to get your hand inside the glove in time to make the catch.
Among the (many!) related ideas, I think that https://patents.google.com/patent/US20070113321A1
has the most practical implementation. It doesn’t look great, though…