If an instruction moves pieces in multiple lands, (e.g., "Unbearable Gaze" pushing a piece in origin land and a piece in target land), can it move the same piece twice: once from the first land to second land, and then from the second land to its final destination?

If an instruction moves pieces in multiple lands, (e.g., "Unbearable Gaze" pushing a piece in origin land and a piece in target land), can it move the same piece twice: once from the first land to second land, and then from the second land to its final destination?

No. "Unbearable Gaze" pushes once and moves two different pieces; it doesn't push twice.
A single push/gather/move instruction will usually not let you move a piece more than once. You are free to move a piece multiple times in so long is it's with different instructions in one action (e.g., Thunderspeaker using "Gather the Warriors" to first gather Dahan with one instruction and then push them with a second instruction) or different actions on the same card (e.g., different players moving the same Explorers with the "Retreat!" Fear Card).