Continuous legal moves given by the player as checks could be 1,2 or more. The checks are not counted as moves. Once the checks are completed, the player will make a single or two moves depending upon his turn which are not check moves.
If I'm understanding this correctly, this seems very degenerate and an easy win for white after 1. e3. No matter what black does for their two moves, it white should be able to play either 2. Qf3 & Qxf7+ OR 2. Qh5 & Qxf7+ OR 2. Qg4 & Qxd7+. Checks don't count as moves, so it is still white's turn. So, white's queen simply gobbles up black's entire queenside with checks: Qxe7, Qxd8, Qxc8, Qxb8, Qxa8. A further continuation (still on white's move) could be Qd8 Qe7 Qf7 Qxf8 Qxg8 Qxh8, gobbling up the entire kingside. Since queens are so powerful (due to the checks aren't moves rule) and black no longer has one (and is also down massive amounts of material), white should be able to win with minimal effort from here.