Prepare the White BBQ Sauce: In a medium mixing bowl, combine all the ingredients for the white BBQ sauce. Whisk until smooth and well combined. Refrigerate the sauce until ready to use.
Cook the Chicken: Heat the vegetable oil in a large sauté pan over medium heat. Season the chicken tenders with salt and pepper, then cook in the pan until fully cooked, about 5-7 minutes per side. Add the sliced shallots halfway through cooking the chicken, allowing them to soften and become translucent.
Prepare the Filling: Once the chicken tenders are cooked, allow them to cool slightly until they can be handled. Chop the chicken into bite-sized pieces and place them in a large mixing bowl. Add the cooked shallots, shredded Monterey Jack cheese, and about 1/2 cup of the white BBQ sauce. Mix everything together until well combined.
Assemble the Subs: Slice the French bread subs down the middle, being careful not to cut all the way through. Stuff the chicken mixture evenly into the subs. Sprinkle a little more shredded cheese on top of the chicken mixture.
Melt the Cheese: Place the assembled subs on a baking sheet and place them under the broiler for a few minutes until the cheese is melted and bubbly. Keep a close eye on them to prevent burning.
Serve: Remove the subs from the broiler and drizzle with additional white BBQ sauce if desired. Serve immediately with extra sauce on the side for dipping.