Make the Garlic Aioli
Mix together mayo, minced garlic, and a pinch of salt. Refrigerate while you prep everything else so the flavors marry up. Taste and adjust garlic if needed.
Prep the Balsamic Lettuce
Toss shredded lettuce with balsamic vinegar and olive oil. Refrigerate.
Cook the Beef
Pound the sirloin thin, then slice into small strips. Heat oil and butter in a pan over medium-high heat. Cook the beef strips, seasoning generously with Montreal steak seasoning while cooking. Remove meat when done and keep warm.
Make the Au Jus
Keep the pan with the beef drippings on medium heat. Pour in beef broth (preferred) or water (about 1 cup). Scrape up all the browned bits from the bottom of the pan with a wooden spoon. Let it simmer for 2-3 minutes to reduce slightly. Taste and adjust seasoning if needed. Strain if desired. Keep warm.
Warm the Bread
Preheat oven to 350°F. Split French bread loaf and place in oven to warm through, about 5 minutes.
Dip and Toast the Bread
Remove bread from oven. Quickly dip the cut faces of both halves into the au jus for just a moment. Place bread open-face up on a baking sheet. Switch oven to broil and broil for 1-2 minutes until bread is toasted and crisping up.
Add Meat and Cheese
Remove bread from oven. On one half, pile on the cooked beef strips and top with half the provolone cheese. On the other half, add the remaining provolone cheese. Return to broiler and broil until cheese is melty and starting to brown (or crisp if that's your preference).
Build the Sandwich
Remove from oven. On the cheese-only side, spread garlic aioli generously. Top with sliced onion and sliced tomato. Season with salt and pepper. On the meat side, top with the balsamic-dressed lettuce.
Combine and Serve
Press the two halves together. Slice into portions. Serve immediately with extra au jus on the side for dipping if desired.