Ace splits on most live blackjack tables receive exactly one card on each ace. You do not hit again, and you do not stand in the ordinary way, because the hand is already finished once that single card lands. Two starting aces become two separate one-card draws. A ten on one of them is a 21, and on many felts that 21 is not a natural, so it pays 1 to 1 rather than the blackjack price. The original pair of aces is not one hand you improve. It is two short hands the rule refuses to let you build.
You do not build a second hand
This is a felt rule. Some rooms print an exception, and the exception is the only reason to take a second card. A chart that says hit or double after splitting aces is describing a table you are not sitting at if the felt forbids it. Players who click hit out of habit on a ten-value upcard discover the button is gone. The decision was made when the aces were split. The dealer pitches one card to each and moves on.
The card count is a felt rule
Read the split-aces line before the first hand, next to the natural payout and the double-after-split line. If the felt allows one card only, split the aces when the chart for that rule says to split, then stop. Do not plan a soft total you will never be allowed to play. If the felt allows more cards, the game is different and the one-card habit becomes the mistake. The shoe, continuous or held, does not change the sentence printed under the betting circles.