Two chess variants where players make two moves per turn are well known: Marseillais Chess, and Double Move Chess.
At an evening in December 1996, celebrating the 50th anniversary of the chess club of Bennekom (a village in the Netherlands), another variant of the same type was played. I didn't see this variant mentioned in Pritchard's Encyclopedia of Chess Variants, and do not know of it being mentioned anywhere else, and hence took the liberty to give the variant the name Bennekom Double Move Chess. It might well be an invention of one of the organizers of this chess variant evening.
The game is played as orthodox chess, but now each turn a player makes two moves with the same piece. (All `double move' variants, I had known until then allowed the player to move different pieces.)
To be precise, a player moves a piece, and when possible, he must make another move with the same piece. When such a second move with this piece is not possible, the second part of the turn is skipped. (For instance, when white has a king on a1 and a pawn on a2, and black has a king on a8 and a pawn on a4, white can move his pawn from a2 to a3 and skip the second part of the turn.)
When a player gives check with his first move in his turn, the second part of the turn is skipped. (Thus, one cannot take a king with two moves in a row.) It is also allowed to give mate with the first move of a turn.
Check should be parried with the first move of a turn.
It is allowed to take two pieces within one turn.