1. To make the dressing: Heat the oil in a medium saucepan over medium-low heat. Add the garlic and cook, stirring, until golden brown, about 2 minutes. Turn the heat to medium-high and add the shallots, carrot, pepper, and chile. Cook, stirring, until the vegetables are tender, about 3 minutes.

2. Add the lime juice, vinegar, salt, and sugars. Turn the heat to low and simmer, uncovered, stirring occasionally, until the mixture is syrupy, about 30 minutes. Almost all of the liquid will have evaporated. Transfer the mixture to a blender. With the blender running, add the chili oil. Blend until completely smooth. Transfer to a shallow dish to cool and set aside.

3. To make the salad: Mix together the soy sauce, vinegar, and sugar in a large mixing bowl. Add the rice flour and ¼ cup water and stir; then add another ¼ cup water and stir. You should have a smooth, lump-free, medium-thick consistency of batter that runs from the spoon but is not completely liquid. Add more water if needed, and stir.

4. Pour oil to a depth of 2 inches in a heavy, deep saucepan and heat to 375°F. Dip the squid in the batter and transfer to the hot oil. Do not overcrowd the pan; work in batches if necessary. Cook until golden brown and really crisp, about 2 minutes. Drain on paper towels and season with salt.

5. Put the papaya, water chestnuts, cashews, ginger, and frisée in a large mixing bowl. Season to taste with salt and toss with the reserved dressing. When everything is well coated, transfer to the serving plates. Top with the fried squid, sprinkle on the sesame seeds, and serve immediately.