1. Preheat oven to 350 degrees Fahrenheit (180 degrees Celsius).
2. Cut the puff pastry sheet in half. Place each half on a Gefen Parchment-lined baking sheet.
3. Create a border by slicing into the edge of the puff pastry about half an inch wide, but not cutting all the way through. Prick the inside of the puff pastry all over with a fork and par-bake for 10 minutes.
4. While the puff pastry is baking, prepare the toppings.
5. For the chicken-broccoli layer, combine olive oil, white wine, hot sauce, garlic, salt, pepper, and thyme in a medium-sized bowl. Toss in broccoli and chicken slivers and toss to coat completely. Set aside.
6. For the mushroom-shallot layer, heat two tablespoons olive oil in a large frying pan. Sauté the shallots for three minutes. Add the mushrooms and season with salt, pepper, and thyme. Continue to sauté for seven more minutes. Pour in teriyaki sauce and white wine and raise the heat. Stir until liquid is absorbed. Remove from heat.
7. Combine garlic-mayo ingredients and spread over the two par-baked crusts, reserving some of the garlic-mayo for drizzling.
8. Divide the mushroom-shallot mixture evenly between the two crusts. Next, sprinkle with marinated chicken slivers and broccoli. Top with a drizzle of garlic mayo.
9. Bake for 30 minutes, or until chicken is ready.