Whisk ½ cup creamy peanut butter, 2 Tbsp. chili sauce (such as sriracha), 2 Tbsp. distilled white vinegar, 2 Tbsp. oyster sauce, 1 Tbsp. fish sauce, 1 Tbsp. sugar, and ¼ cup water in a medium bowl until smooth and creamy. If sauce is thick, add a few splashes of water to loosen to the consistency of ketchup.
Toss 2 cups shredded chicken meat (from about ½ small rotisserie chicken), 1 Tbsp. chili sauce (such as sriracha), 1 Tbsp. oyster sauce, and a pinch of kosher salt in a medium bowl to coat chicken.
Working one at a time, dip twelve 8"-diameter rice papers into a large shallow bowl filled with cool water, just long enough to coat in a film of water (you don’t want them to be floppy). Shake off excess water and place on surface. Place 1 leaf (or 2 if leaves are small) from 1 head of butter lettuce, leaves separated, on bottom third of a paper to act as a cup. Arrange about ¼ cup of 3 cups 2" matchstick crunchy vegetables (such as carrots, radishes, and/or cucumbers), 2 Tbsp. seasoned chicken, a few pieces from 1 large mango, peeled, pitted, cut into 2" matchsticks, and a few pieces from ½ bunch cilantro, tough stems removed, cut into 1" pieces, on top of lettuce. Fold sides of rice paper up and over filling, then starting with the side closest to you, roll up rice paper gently but tightly. Repeat with remaining ingredients to make 11 more rolls.
Lightly coat a platter with vegetable oil and arrange summer rolls on top. Cut in half diagonally if desired and serve with peanut sauce for dipping.
