I know you shed a chain when drawing your initial hand at the start of a game but if you mulligan do you lose an additional chain?

You will lose one when you go to draw at the end of the turn.

Mulligan, Draw 5 and keep, shed one chain, play your card for the turn, draw a card up to 5 and shed a chain.

First player will lose two chains during their first turn 99% of the time.

EDIT - Rereading your question, you don't lose the first chain until you keep your hand. So you don't draw, lose a chain, mulligan and lose another chain.

Thx