Last time I tried to cook sunny side up eggs on a metal skillet. I put butter on the skillet, allowed it to heat and then I cracked two eggs on it. Immediately I placed a glass lid with a hole on the eggs.  I found that the eggs did NOT turn "brown" from below (I kept on checking them periodically by lifting them up by a spoon). After waiting for quite some time I saw the egg yolk wearing a white layer on them (from the top). I turned off the gas, and to my horror those were the most "hard" eggs I had ever eaten.  Without a thermometer what is the way to know that the "time's up"!  P.S. I DO want the salmonella to be killed. 
The reason, I think, that most recipes don't leave a thick caramel and it all goes 'runny' is that it's designed to go into a sort of caramel-y sauce when turned out.  Since in most conventional recipes you par-cook the fruit in the caramel then top with the pastry and bake, the fruit releases a lot of moisture into the caramel and making it runnier. Therefore, if you cook the apples separately (boiling them, say) then drain, add to the caramel and cover with pastry and bake, much less juice should leak out of them.  I would keep the caramel recipe the same as in the recipe but just follow the steps above. I don't think the type of sugar will make any difference other then flavor (although soft brown sugars do have more moisture in them).  Also, since everything except the pastry is cooked, try baking the pastry conventionally (in the oven) then grill it for a few minutes to increase the heat on the pastry directly to avoid it becoming soggy. 