Preheat oven to 375 F
Put the bread mix into a large bowl.
In a separate bowl, beat the egg white and mix it in with the water and vinegar until combined.
Pour the liquids into the bowl with the bread mix and stir well to combine. Knead with your hands until it comes together as a wet dough.
Place the dough onto a piece of parchment paper and top with another parchment. Roll it very thin in between the paper. You want it to be about 1/8th of an inch thin and it should be about 16 in. by 12 in wide. Remove the top piece of paper and put the dough (along with the bottom parchment) onto a half-sheet pan. Use a fork to poke small holes throughout the dough, this will prevent the middle of the dough from puffing up.
Bake for 25 minutes.
Meanwhile, put the pesto ingredients into a food processor and process until smooth.
Remove the flatbread from the oven and press it down a little in the center. Spread on the pesto, followed by the chicken, and then the red pepper slices.
Put the flatbread back into the oven and cook 10 minutes until the toppings are warmed through.
Slice and serve.