1. Combine vinegar, 1/4 cup water, 1 teaspoon salt, sugar, and thyme in a saucepan. Bring to a boil, stirring until salt and sugar dissolve. Pour hot mixture over squash in a bowl. Let stand for 30 minutes, tossing occasionally. Drain.
2. Combine mayonnaise and next 4 ingredients (through almonds) in a mini food processor; process until smooth. Set sauce aside.
3. Heat a grill pan over medium-high heat. Sprinkle chicken evenly with remaining 1/4 teaspoon salt and black pepper. Coat pan with cooking spray. Add chicken to pan; cook 4 minutes on each side or until done.
4. Spread about 2 tablespoons sauce on bottom half of each roll. Top each with one chicken cutlet, about 1/3 cup squash, 1/2 cup arugula, and top half of roll.