Color the unit cells of the square in checkered fashion, with top-leftmost cell black. There are one more black cells than white. If we cut a $11\times 11$ square with top-leftmost cell white, there will remain two more black cells than white, and since any domino covers exactly the same black and white area, the requirement would be impossible to fulfill. If however, we cut a $11\times 11$ square with top-leftmost cell black, there will remain a same number of black cells as white, and the requirement will easily be fulfilled (fill pairs of rows/columns, then borders). There are therefore $2\cdot 1000^2 + 2\cdot 1000 + 1 = 2,002,001$ ways.