Reply to "Dough Breaks in Thin or Detailed Areas Even When I Carefully Lift Up"

First, I use parchment paper and plastic wrap on top.  Second, try wiggling the cookie cutter before you remove the cutter.  The more times you use the cutter the better it will work.  May i suggest using a combination of flour and powdered sugar to roll your cookies.  They will dry out less and you will feel comfortable using more of the combo when you are flouring your cutter. God luck!