ROAST THE RED BELL PEPPER: Preheat the broiler. Place the red bell pepper halves on an aluminum-foil-lined rimmed baking sheet. Place under the broiler and roast until blackened, approximately 10 minutes. Fold the aluminum foil over the roasted pepper and let it steam for 10 minutes. Then, peel off the blackened skin and cut each half into 3 long strips.
CHAR THE ZUCCHINI AND YELLOW SQUASH: Heat a nonstick skillet over medium-high heat. Dry fry the zucchini and yellow squash in batches, turning once after 1-2 minutes. Cook until somewhat tender but still crisp. Adjust the heat as necessary so the skillet doesn't smoke. (You might need to go lower and slower at times.) Transfer to a plate when each batch is charred and season with salt and black pepper.
COOK THE SHALLOT: Heat the olive oil in the nonstick skillet over medium-low heat. Add the shallot and cook for 2-3 minutes or until softened, being careful not to brown it. Add the garlic, cook for 5-10 seconds, then remove from the heat. Tip the skillet to let any excess oil drain off so the shallot and garlic can cool.
BEGIN ASSEMBLY: Unfold the puff pastry onto a piece of parchment paper large enough to fit on a rimmed baking sheet. Using a little all-purpose flour on the rolling pin if needed, roll it out to a thickness of 1/4 inch. Transfer the pastry on the parchment paper to a baking sheet.
Spread the basil pesto over the entire pastry, leaving a 1/2-inch to 1-inch border.
Mentally divide the pastry lengthwise into three sections. On the right 1/3 of the pastry, drop the goat cheese in pieces. (It will look overloaded, but you'll take some out later.
Sprinkle the mozzarella cheese on the center 1/3 of the pastry.
Spread the shallot-garlic mixture over the mozzarella.
Arrange the vegetables over the center portion only as follows:-- 1/2 the zucchini, 1/2 the yellow squash, and all of the red bell pepper.--Sprinkle the vegetables and goat cheese with 1/2 of the Italian seasoning.--Remove some of the goat cheese from the right section and place it in the center of the red bell pepper.--Top with the remaining zucchini and yellow squash. Sprinkle with the remaining Italian seasoning. Beat the egg with the water. Using a pastry brush, coat the edges of the puff pastry with the egg mixture.
Press the goat cheese down onto the pesto. Carefully and quickly, using the parchment paper to assist, fold the right panel (the one with the goat cheese) over the vegetables in the center panel. (It should cover at least half of the vegetables.)
Using the parchment paper to assist, flip the left panel with just the pesto over the middle. Press the edge onto the strudel to form a tight seal. Then, carefully flip the strudel so the seam is on the bottom, tightly seal the top and bottom, and fold them under as best you can, ensuring a tight seal with the beaten egg. (This helps keep the strudel tightly sealed and gives it a smooth, attractive appearance.)
Decoratively cut three shallow slits in the top to let steam escape.
Keep the strudel on the parchment paper, cover with plastic wrap, and chill in the refrigerator for 1 hour or in the freezer for 20 minutes.
Just before baking, brush with beaten egg.
Preheat the oven to 400 degrees. Bake for 40-45 minutes or until golden brown. Let rest for 10 minutes before slicing. Serve warm.