How to Make It
Halve cucumbers lengthwise. Cut into 1½-inch lengths. Place one-third of the cucumbers in a gallon-size ziplock plastic bag. Smash lightly with a meat tenderizer or saucepan just to break the skin. (Don't pulverize them.) Transfer smashed cucumbers to a large bowl. Repeat process with remaining cucumbers.
Add granulated sugar and 1½ teaspoons of the salt to cucumbers, and toss to coat. Pour cucumbers into a strainer set over a bowl. Refrigerate 1 to 2 hours.
Combine fish sauce, lime juice, brown sugar, ginger, and chiles in a bowl; whisk into a paste. Let stand 15 minutes to allow flavors to bloom. Whisk in sesame oil and canola oil.
Remove cucumbers from refrigerator, and discard any accumulated juices. Transfer cucumbers to a large bowl. Add sprouts, carrots, cilantro, mint, ¾ cup of the peanuts, and remaining 1½ teaspoons salt. Drizzle with dressing; toss to coat. Sprinkle with remaining ¼ cup peanuts, and serve.