When pouring a drink (generally shots), and the consumer of the said drink doesn't want a full shot, but a little more than half, he or she would ask for generous halvises.
"Don't give me a full shot, just give me half!"
"Bitch, you're taking a generous halvsies shot!"
When you can't decide between two things, so you and another person (usually a friend or acquaintance) each purchase one of the desired items and you both exchange half of each item with one another.
Note: Most often done when ordering food. This way you can eat both items you want instead of ordering two meals and looking like a fat ass.
Person 1: "Man, I can't decide between the teriyaki chicken or the club."
Person 2: "Yeah, me neither."
Person 1: "Double halfsies?"
Person 2: "You read my mind."