Go for the quick kill or grind them down when you are gifted.
I personally do the opposite when I benefit from a blunder. I attempt to make the position as calm as possible, trade down, and focus on setting myself up an easy to convert end game. Attaching my bishop to a pawn at the base of a pawn chain is a good example. One of my first thoughts after I gain a large advantage is which pawn am I going to eventually try to promote. You can definitely take this too far if you're advantage is not that large, and don't let this philosophy prevent you from capitalizing on your opponents next mistake.
It always depends on the position. Sometimes a blunder can open up your opponent to a crushing attack. Other times, the blunder may put you in a good position but still leave your opponent with decent attacking or defending chances. In that case, you need to grind them down.
I'm a 700 ELO who plays 10 Minute Rapid. From my experience, persistent checking in hopes of landing into a mate usually just leads to my opponents catching up through free development or traps. Even while holding multiple pieces of advantage, I always focus on continuing to build that advantage while making even trades if need be. This often leads to me missing quick mates, but these seldom influence the outcome of the game. Taking the time to analyze every position for potential mates can waste time compared to playing defensively and waiting for endgame.
After a blunder, take a moment to evaluate the best way to capitalize on it. Consider both immediate tactical opportunities and long-term positional advantages.
I'm going to be perfectly honest with you, it depends on what the advantage is (material advantage, open files, outposts, etc.) and it depends on how big the advantage is. If i have like a rook up or better or simply a potential big attack ill usually attack, if not, defensively play until the endgame where you convert a winning endgame and win. simple. (pawn(s) or minor piece up.)