Prepare the pork: Mash the garlic into a paste with the flat side of a knife. Transfer to a large bowl and stir in the lemongrass, shallots, brown sugar, fish sauce, soy sauce, vegetable oil and 1/2 teaspoon each salt and pepper. Add the pork and toss to coat. Cover and refrigerate 2 hours or overnight.
Make the sauce: Whisk the lime juice, fish sauce, granulated sugar, 1/2 teaspoon salt and 1/4 cup water in a small bowl. Stir in the carrot, chile sauce and garlic.
Preheat a grill to high. Brush the grill with vegetable oil. Remove the pork from the marinade and grill until well marked, 3 to 4 minutes per side. Transfer to a cutting board and let rest 5 minutes, then cut into strips. Arrange the pork and lettuce on a platter; set out the herbs, cucumber, jalapeno and fried shallots for fixings. Assemble the lettuce wraps and serve with the carrot sauce and shrimp chips.