Question for the group. It seems logical to me that someone should both make their move and notate it (and let’s assume for my purposes that we are not using the paper scoresheet variation that allows one to write the move down before making it) before hitting the clock. My personal observation is that players often make the move, hit the clock, and then “write it down” (either on paper or electronically). Thoughts on whether or not this is proper? If it is not, can someone point me to specific language in the rulebook that states that you must write the move down before hitting the clock?
There is absolutely no requirement that the player must record his move while his own clock is running. It is both acceptable and customary to execute the move on the board, press the clock, and then record the move.
Sometimes (especially if the reply to an opponent’s move is obvious, such as a recapture), you will see a player execute a move, press the clock, and then record both the opponent’s move and his own. This, too, is acceptable.
FIDE hasn’t thought of that ‘innovation’ yet, apparently. 
How many moves or move pairs can I get away with by not writing them down? A player can skip writing down a memorized opening for a number of moves and wait until an opponent starts thinking so that he can then write down the moves. This could be a nice advantage in both delay and increment time controls. Any skipping of writing of the move borders on a lack of sportsmanship, never mind not following the rules.
When moving somewhat quickly I generally move, hit the clock, write my opponent’s and my move. If I am spending a fair amount of time on my I’ll write my opponent’s move on my time and still write mine after hitting the clock.
Rule 15 says that you have to keep score move after move. Some define move as meaning the individual white and black moves (arguing that you have to write down your opponent’s move before making your own) and others argue that move and move pair are interchangeable (arguing that you can write down both players’ moves at the same time).
Since the definition of a complete scoresheet is missing no more than three move pairs, waiting until after your move to write down your opponent’s move means that the scoresheet might at worst be considered incomplete to the tune of one more move pair (even if only one player’s move was missed the move pairs are still counted as integers, not in halves). So if somebody is getting very picky about specifically when each player’s move is written I simply fall back on the allowable incompleteness of a scoresheet for making claims.
I have been told by IAs that writing both moves at the same time is acceptable to FIDE.
So, I guess that means in USCF events I can skip two move pairs while making forced captures or making two quick moves in the opening or endgame to gain time, and then go back and fill in the moves while my opponent is thinking. In FIDE events I cannot do that and must write the moves down. There is a certain degree of gamesmanship that can stretch the rules. Some players rush moves and bang the clock to get opponents distracted and out of rhythm.