No experience with lamb here, but in my experience the soft cheese is
most likely the culprit. With low melting points, some of those
cheeses like feta, bleu, and the mexican soft cheeses simply melt out
of the patty when they can.

I have had much better luck when I add the cheese after the the flip
like they do at my favorite burger joint.

As always, just my 0.02.

Robert