Using a Japanese mandolin or a sharp knife, thinly slice the cucumbers about 1/16-inch thick. Place the sliced cucumbers in a large colander and toss well with 1 teaspoon kosher salt. Place the colander over a large bowl and allow the cucumbers to sit and begin to drain for 10 minutes.
Meanwhile, whisk together the sour cream, white balsamic vinegar, olive oil, mint, chives and the remaining 1/8 teaspoon salt in a large bowl. Set aside.
Using your hands or a rubber spatula, squeeze or press out the excess water from the cucumbers. It is best to work in small batches to extract as much water as possible. Add the squeezed cucumbers to the sour cream dressing. Using a rubber spatula, toss the cucumber in the dressing until well coated, and serve.