Preheat oven to 350º F and set up the middle rack.
Combine the vegetables in a large bowl; drizzle with 1 TBSP of olive oil, season with salt and pepper, and toss to coat. Set aside.
Place the cut up chicken in a separate large bowl; drizzle with 1 TBSP of olive oil, add garlic, and season with salt and pepper. Toss to coat.
Evenly spread the vegetables on one side of a baking sheet, and the cut up chicken on the other side. Place in the oven and bake for 10-12 minutes, or until veggies are tender.
Remove the sheet pan from the oven and set aside.
Slice each hoagie roll down the middle just enough to open the roll, but don't cut it all the way through. Butter the insides of the hoagie rolls. Carefully slide the veggies and chicken over to make room for the rolls, and set them down on the sheet pan, buttered side up. Place slices of provolone evenly over the chicken.
Set the broiler to high, and place the sheet pan into the oven. Let the ingredients broil for 3-5 minutes, or until the hoagies are toasted golden and the cheese has melted. (Every broiler differs, so be sure to keep a close eye on things so they don't burn.)
Remove the sheet pan from the oven. With a spatula, carefully stuff each hoagie with the veggies and cheesy chicken.
Slather mayo onto your hoagies if you're into it. (We are!) Enjoy :)