Since a single line doesn't do damage, it is possible to do
To achieve this, there are a couple of requirements:
To get these pieces one after the other
For this to occur so that the final piece also clears the board, we get these constraints on the number of pieces $X$:
Given these, the smallest $X$ that satisfies both requirements is
This is a small ...
Unless I made a mistake somewhere, this solution is unique:
To get there, I used a couple of rules of thumb:
Apart from those, there were a couple of slightly mind-boggly deductions required, but all in all, everything seemed extremely well designed, and no guesswork was needed at any point.
Progress, part 1:
Progress, part 2:
Progress, part 3:
@Daniel_Mathias gave a very helpful link which has all the 12x5 solutions in a text file. So some simple code allows us to see that of the 1010 12x5 solutions, there are 264 with 1 straight cut. But, sadly, none with 2 or more cuts. A few examples of the former are: