My english is okay, but not good enough to describe this, so I made a picture.
This is what happens in our real life (boring) $3$D world,
Note that if we slice the apple one more time (unless you slice again, parallel to the earlier cut), there's no way of putting your apple in the table having the face(s) of the region that's been cut touching the table, and the rest (the curvy parts) not touching it.
Is it possible to make more cuts like the one showed while still being able to achieve the condition above in more than $3$ dimensions?
PS: Please ask for clarification if needed, this is kind of difficult to explain (especially not using my native language).
Approximate an apple in i.e. $D=4$ as a unit sphere of radius $R$, where the outer shell is defined by the coordinates
$$\sum_{i=1}^4 x_i^2=R^2$$
If we are talking about a table (as used by humans) then we are basically considering a $D=2$ surface where objects can be placed upon.
After performing the first cut, orient your coordinate system such that the cut surface coincides with table surface - the sphere (apple) is put on the table with the cut surface downwards.
By the remaining $SO(3)$ rotational symmetry of the cut sphere (rotation axis through the original center of the sphere, and normal to and through the center of the surface of the cut), we can project the shape onto a $D=3$ cut sphere without losing information. Furthermore, the $D=3$ cut sphere still has an $SO(2)$ symmetry (same rotational axis), so we can proceed to project it onto a cut $D=2$ sphere (cut off circle) without losing information.
As we try to apply the second cut, we can always orient our double projections above such that the second cut appears as a line in the cut circle. Therefore, whether we start in $D=3$ or $D=4$ or general $D\in\mathbb{N},D>2$ to apply the initial cut, by symmetry the problem reduces to just a cut circle and the result is the same:
You cannot cut a $D$ dimensional sphere twice and put it on a table such that no cut surface can be seen, unless your second cut removes all information about the first cut from the remaining shape.