The OED has cheese-cutter with a hyphen. If you like that style, you might prefer to write cheese-slicer.
On the other hand, the Wikipedia article on Cheese knife uses both cheese cutter and cheese slicer uncluttered by hyphens. They present this as an example of a cheese slicer, which they label as such:
Personally, I wouldn’t write them as one word with neither hyphen nor space separating them, but that’s just me. The British do tend to use more hyphens than North Americans, but to my eye that’s just unnecessary clutter in many cases (like nonaligned, for example).