Alex Y
Place the torus centered at the origin, with its major diameter on the x-z plane
Cut 1: The y-z plane, leaving you with two u-shaped halves. Call them A and B.

Cut 2: Rotate the cutting plane 20 degrees around the z axis, cutting a little wedge off each of the four ends of the Us, for a total of six pieces.Call the new wedges A1, A2, B1, and B2, and A and B refer to what is left of the U's.
Cut 3: rotate the cutting plane an additional 10 degrees around the z axis, and also 5 degrees around the y axis.

If I am thinking about this correctly, cut three will take a new small wedge off of each end of A and B (four more pieces) and will divide one of the wedges from each half, e.g. A1 and B2 or A2 and B1.

I think I need to go buy a bagel and try this, because I sure have a hard time visualizing what I am saying, so have very little confidence of this answer!