Tell Burke to never hire Carlyle, I forget the specific coach but I think they were in on Babcock back in 2013 before they hired Carlyle.
In the 2009 free agency avoid Komisarek and Beauchemin, snag Boychuk, Seidenberg, or even Greene at all costs. Beg Burke to sign one of Cole, Moen or Moulson to shore up the forward corps. Jettison Toskala by any means necessary and pickup Anderson.
Go into the season with a lineup of:
Moulson - Stajan - Kessel
Blake - Bozak - Stempniak
Kulemin - Tlusty - Stalberg
Orr - Irwin - Deveaux
Kaberle - Boychuk
Schenn - Greene
Gunnarsson - Finger
Anderson
Gustavsson
While not amazing, there isn't many impact forawrds to be had in the draft. Moulson would go on to put up 30 goals that year with NYI. With most of the money spent on the backend, the Leafs readjust their big contracts to Komisarek and Beauchemin and sign a talented puck mover in Andy Greene who averaged ~25minutes a night on the ice and racked up 31 assists on the Devils blueline. This allows Schenn to focus on his defensive game and not get caught with the grenades Beauchemin/Komi can throw. The Leafs have an effective breakout guy for Wilson to utilize. Meanwhile, Kaberle gets a steady defender with a great right handed shot that can shut down top lines and the Leafs get a guy who continues to be a top level defender up to today. Leafs pick up Anderson just in time for his cinderella season including his huge playoff performances.
2010 Trade Deadline: Interfere with division rivals and interrupt the Cam Barker trade, getting Kim Johnsson and Nick Leddy from the Wild instead.
I also forgot to list White, Stalberg and some others in this, hard to remember who was around for the start of the season, but this gives the Leafs a decent defensive group to help their scorers get into the offensive zone with a fast paced Wilson system helping them pry themselves out of an abysmal East and save themselves the embarassment of giving the #2 pick, leaving underwhelming players like Gudbranson, Nieterriter, Skinner, McIlrath, and Connolly.
Advise Burkie instead of taking Brad Ross the next Darcy Tucker
instead take the big truculent 6'3 winger from the Ottawa 67's, Tyler Toffoli. Avoid Daniel Brodin with the 146th pick and instead take the 147th pick, Brendan Gallgher.
Keep the 2010 free agency the same, it was largely a hit for the Leafs with the addition of MacArthur, could have done without Colby Armstrong though, snag Antti Niemi from Chicago instead of letting him go to San Jose.
2011 draft, advise Burke not to take the prototype power forward in Biggs but instead the touted defensive forward in Saad.
2012 trade deadline season: Sell high on Joffrey Lupul, yea he's an All-Star, but you have Brandon Saad, he'll anchor that line and in no time you'll have JVR. Get rid of soft shooters and double down on truculence. His short contract will only help his price for playoff desperate teams. Columbus being the West needs to move their disgruntled Center, why not send him East? Toronto offers Phaneuf and a pick besting LA's offer of Jack Johnson and a conditional 1st to get Carter.
2012 draft: What more could you criticize? Garbage draft, but Leafs walked away with one of the best. Maybe Lupul could have been traded and turned into a package to get Filip Forsberg before the Caps scooped him up at 11th.
2012 Free Agency: Keep the status quo, the only impact player that could have helped might have been Jagr, using his work in Philly to pitch to Burke how good he help the youngsters of Toronto wiht his work ethic and his steady handedness.
2013 season: Kill Dave Nonis. Just do it. In February, with their renewed offense and a solid back-end, the Leafs replace Phaneuf with Regehr, sending picks and potentially some prospects Buffalo's way. Besting LAK's 2 2nd round pick combos (used it for Bolland, so why not Regehr to fill in that 2nd pairing spot?)
2013 Draft: Keep it Gauthier unless people really think Poirer, Dano, or Shinkaruk are amazing. Alternatively, use it to trade for Robyn Regehr.
2013 Free Agency: Let Bozak go keep KGM together, probably as they slump down to 3rd line behind:
JVR - Carter - Kessel
Saad - Kadri - Toffoli
Kulemin - Grabovski - MacArthur
while keeping:
Rielly - Boychuk
Leddy - Regehr
Gunnarsson - Fraser
Niemi
Reimer
Its not perfect, but you trade the Leafs defensive hassles for longterm solution while keeping a solid offense intact that at its worse is thin down the center, but you still manage to keep a high-scoring, balanced offense running through three lines, and in the end you maintain Burkie's vision of hard to play against big bodied players and a solid goaltending/defensive group. Possible deals to try and interject into at the deadline: Blake Wheeler+Mark Stuart for Rich Peverley+Boris Valabik. Wheeler, even though the Leafs have Toffoli, would be the Leafs 2nd line RW, slotting in behind Kessel. Bishop for 2nd round pick swap in 2012 would give the Leafs a big backup to go behind Niemi, if you don't think Reimer would be the answer.