I am shamelessly stealing this idea from Royals Review, where they did it for the Central. I used http://www.mlbdepthcharts.com/ and Fangraphs to figure everything out. The position is given along with the projected starters on each team, and the winner at that position, the player with the highest projected WAR (CHONE).

Left Field – Randy Winn, Jacoby Ellsbury, Noland Reimold, Travis Snider, Carl Crawford. 
Jacoby wins with 3.9 WAR

Center Field – Curtis Granderson, Mike Cameron, Adam Jones, Vernon Wells, B.J. Upton
Granderson wins with 4.4 WAR (CHONE puts Cameron at 1.7 even though he had two straight 4+ years, seemed very odd to me)

Right Field – Nick Swisher, J.D. Drew, Nick Markakis, Jose Bautista, Matt Joyce
Markakis wins with 4.1 WAR (CHONE puts Drew at 2.7 despite two straight 4+ years, another oddity)

Third Base – Alex Rodriguez, Adrian Beltre, Miguel Tejada, Edwin Encarnacion, Evan Longoria
Longoria just gets it with 5.1 WAR (Rodriguez was at 5.0)

Shortstop – Derek Jeter, Marco Scutaro, Cesar Izturis, Alex Gonzalez, Jason Bartlett
Jeter wins with 3.4 WAR

Second Base – Robinson Cano, Dustin Pedroia, Brian Roberts, Aaron Hill, Ben Zobrist
Pedroia wins with 4.7 WAR (Zobrist had a fairly conservative 3.7 after his breakout)

First Base – Mark Teixiera, Kevin Youkilis, Garret Atkins, Lyle Overbay, Carlos Pena
Teixiera wins with 5.1 WAR (Youkilis posted consecutive seasons of 5.5 and 5.6, but CHONE has him at just 3.8)

Catcher – Jorge Posada, Victor Martinez, Matt Weiters, John Buck, Kelly Shoppach
Martinez takes the last spot with 4.4 WAR

It all adds up to a 35.1 WAR for the All-AL East team.