Installing thin slate (chalkboards) as countertops: proper method?
I would handle this as a tile countertop.  Instead of small 4x4 tiles you will be working with the larger slate sheets.  This means, 3/4 plywood down as a base.  Cement board on top of that and then thin set to attach the tiles.

You can then trim the outsides in whatever you want.  It can be wood, tile or metal.  You can also use the slate cut into smaller pieces to trim as well.

Once everything is set, you should then use grout to fill the gaps between the slate sheets.

Remember, just like tile you need a good solid base; so skip the MDF and use plywood.