Preheat the oven to 350ºF.
Lightly coat a 9x13-inch baking pan with cooking spray.
Unroll one can of refrigerated crescent rolls or crescent sheet dough.
Place the dough in the bottom of the prepared baking pan.
If using crescent rolls with perforations, press the seams together to form one even sheet of dough.
Layer the thinly sliced deli ham evenly over the dough.
Arrange the Swiss cheese slices over the ham.
Layer the thinly sliced salami over the Swiss cheese.
Arrange the provolone cheese slices over the salami.
Layer the thinly sliced pepperoni over the provolone cheese.
Place the drained roasted red pepper strips evenly over the pepperoni.
Unroll the second can of refrigerated crescent rolls or crescent sheet dough.
Place the dough over the roasted red peppers.
Press the seams together, if needed, and gently seal the edges as much as possible.
In a medium bowl, whisk together the eggs, grated Parmesan cheese, and basil pesto until fully combined.
Pour the egg mixture evenly over the top layer of crescent dough.
Cover the baking pan with aluminum foil.
Bake the antipasto squares, covered, for 30 minutes.
Remove the aluminum foil.
Continue baking for an additional 15 to 20 minutes, or until the top is golden brown and the dough is fully cooked.
Remove the pan from the oven.
Allow the antipasto squares to rest for 5 to 10 minutes before slicing.
Cut into 24 squares.
Serve warm or at room temperature.