How to bake lokum rolls well?
I would start by rolling the pastry dough thinner; I often have my dough too thick the first time or two that I work with a recipe.  Keep the dough cold to make it easier to work with.

Next, I would try different oven times and temperatures.  Try a temperature 25-50 degrees (F) higher, and pull the rolls out 30 seconds to one minute sooner.  The nice thing about playing with oven times and temps is that you can go through many tests quickly by just putting a few of the items in for each time/temp combination.  (Extra mass in the oven, such as a pizza stone or even some bricks, will help keep the oven temp more steady.)

If the outside of your rolls are getting over done before the inside is done, try chilling the rolls in the refrigerator for 30 minutes or so before baking.