Make the pork patties: In a large bowl, combine the ground pork, onion, garlic, fish sauce, soy sauce, and sugar. Mix thoroughly. Cover with plastic wrap and refrigerate for 30 minutes to 1 hour.
Form the chilled meat mixture into 8 burger-like patties, weighing around 2 ounces each. Brush the vegetable oil over the patties.
If you’re using a grill: Preheat the grill to high. Grill the patties 4 to 6 minutes per side, until charred and cooked to an internal temperature of 165°F. Transfer the patties to a plate to rest.
If you don’t have a grill: Preheat a 12-inch skillet over medium heat for 2 to 3 minutes. Add the patties to the pan oil-side down and brush the other sides. Fry until browned on the bottom, 2 to 3 minutes. Flip them over and brown the other sides until cooked through to an internal temperature of 165°F, another 2 to 3 minutes.
Make the sauce: In a small bowl, whisk together the fish sauce, ½ cup water, the sugar, vinegar, and juice of 2 or 3 lime wedges. Taste and adjust to your liking with more lime juice, if needed.
To serve: Cook the vermicelli noodles according to the package directions.
On a serving platter, lay down a bed of the lettuce, scallions, and fresh mint. Pile on the vermicelli, cucumber, and pork patties. Serve with the sauce in a small bowl on the side with more lime wedges.
Nestle a few carrot pieces between the noodles and sprinkle some peanuts over each bowl.
