Make the summer rolls: Place noodles in a heatproof baking dish, cover with hot water, soak 8 to 10 minutes, drain, and rinse under cold water.
Place 1/2 inch of cool water in a pie plate. Submerge a spring roll skin in water for 10 seconds. Remove from water and transfer to a clean work surface. Place 4 shrimp halves, cut side up, in a straight row across lower third of spring roll skin. Top shrimp with a few basil leaves, a few tablespoons of shredded cabbage, a pinch of carrots, and 1/3 cup rice noodles.
Carefully lift edge of spring roll skin nearest you up and over filling. Fold sides over filling and continue to roll away from you until edges are sealed. Transfer summer roll to a platter, cover with a damp paper towel, and continue building remaining rolls.
Make the dipping sauce: In a medium bowl, whisk garlic, jalapeno, sugar, lime juice, fish sauce, and scallion to combine. Transfer to dipping bowls and garnish with sliced scallion tops. Serve with summer rolls.