If you don't own a salad spinner, lay the green beans on a clean dish towel to dry in step 2. The blanched, shocked, and dried green beans can be refrigerated in a zipper-lock bag for up to two days.
Total time: 55 minutes
1. Add oil and shallots to 8-inch nonstick skillet and cook over medium-high heat, stirring frequently, until shallots are deep golden brown, 10 to 14 minutes. Using slotted spoon, transfer shallots to paper towel–lined plate; season with salt to taste. Pour 3 tablespoons shallot oil into large bowl and set aside (reserve remaining oil for another use).
2. Bring 2 quarts water to boil in large saucepan over high heat. Add green beans and ¼ cup salt, return to boil, and cook until green beans are bright green and tender, 5 to 8 minutes.
3. While green beans cook, fill large bowl halfway with ice and water. Drain green beans in colander and immediately transfer to ice bath. When green beans are no longer warm to touch, drain in colander and dry thoroughly in salad spinner.
4. Add lime juice, fish sauce, 1 tablespoon water, sugar, paprika, and cayenne to bowl with reserved shallot oil and whisk until sugar is dissolved. Add green beans, carrots, and 2 tablespoons cilantro and toss to combine. Transfer to platter; sprinkle with peanuts, fried shallots, and remaining 2 tablespoons cilantro; and serve.