I freeze my fondant onto the mat, too. I find it cuts very well that way. I also leave it on the mat about 20 minutes before I try to take it off. When it freezes and comes to room temp it becomes very soft. Sometimes the cuts aren't all the way through and when I try to remove the excess fondant around the image it drags it. I leave the image on the mat to air dry for awhile. As I remove the image from the mat I first remove the excess around the cut image. I keep my exacto knife nearby incase the cut didn't go all the way through all the way around. Sometimes, depending on whether or not the fondant is dry enough to remove the excess, I may leave the cut image on the mat a few minutes longer. I then use a small spatula or scraper to remove the image from the mat and set onto parchment paper to dry.
I've also had luck removing the image while frozen as well if I don't have time to wait for it to dry. After I cut the frozen image I gently remove the excess from around the image. If I need to help the cuts with my exacto knife I cut around the image. I then put the mat back into the freezer to firm up before I try to remove it. Once it's very frozen I find it easy to remove. I also use quite a bit of shortening on the mat. No enough shortening to see the white of it, but enough to be able to slide the image off the mat easily.
I've also read others who have left their fondant/gumpaste on the mat to dry a few hours before cutting it. I've never tried that but it may help.
Don't forget to adjust your speed and pressure and make sure your fondant is rolled thinner than what you would use on a cake. I make mine about 1/8 of an inch or less at times depending on what I'm cutting.