Wash and peel raw prawns, leaving the last tail piece on. Devein and cut open lengthwise to butterfly them. Keep refrigerated.
Peel cucumber, preferably creating decorative wave patterns by peeling in stripes. Halve lengthways and slice thinly.
Arrange grated cabbage and carrot mixture in the center of a large plate. Place butterflied prawns in a circle around the vegetables with tails pointing inward. Drizzle with a little fish sauce. Arrange halved cucumber slices around the outer edge. Cover with cling film and refrigerate for at least 1 hour.
In a large heavy mortar, pound chopped garlic, chopped chillies, and salt until beginning to break down. Add chopped coriander and continue pounding for at least 5 minutes until a fine paste forms.
Season the paste with brown sugar, a little fish sauce, salt, and lime juice. The dip should be very spicy, slightly sweet, sour, and not very salty—with a mushy consistency, not liquid.
Serve the prawns and vegetables on the plate. Place the dip in the mortar or a small bowl next to them. Arrange the sliced garlic and sliced chillies as garnishes on the side.
To eat: place some cabbage and carrot mixture on a spoon, top with half a cucumber slice and a butterflied prawn (tail pointing at the tip of the spoon). Top with a slice of garlic (optional), some chili slices (optional), and roughly 1/2 teaspoon or more of dip as desired.