Grilled Halloumi

In our local market, I had seen Halloumi* cheese, a sheep-and-goat's milk cheese that comes from Cyprus, which has the unusual property of a very high melting point, allowing it to be grilled. It sounded intriguing, so I decided to try it. I found this recipe on RecipeZaar, which sounded good. Cut the cheese into cubes, along with grilling veggies (I used green and red bell peppers, onions, and brown mushrooms). Prepare a marinade of fresh herbs (I used fresh thyme for that Mediterranean flavor, with a little prepared ground marjoram), lemon juice, olive oil, garlic, and seasoning, and marinate the veggies and cheese overnight. Then skewer them and grill, about 8 minutes, turning every couple of minutes.

The result was delicious. The veggies were very flavorful, and the cheese has a great taste, something like feta, but with a unique texture, firm and almost squeeky, somewhat like calamari. The pieces that got grill-charred had a nice grilled-cheese taste to them. However, contrary to what I'd read, the cheese does melt if it gets hot enough, and my cheese chunks melted off of their skewers and made a bit of a mess on the grill. I was able to pry much of it off, and what I pried off (even the charred bits) were very good. (The cheese resumes its firmness when it cools.) But it's clear I need to experiment with technique here. Lower temperature? (I had my gas burners on high.) Put them on a higher rack? (I had them on the main lower grill.) Anybody out there have experience with this?

* "Halloumi" is how they spell it in the English translation, but it's spelled with a chi in the Greek,
χαλλουμι, so that initial H is the thick heavy sort produced from the back of the throat.