Bring 2 cups of water to a boil and add ½ teaspoon of salt and the farro. Lower heat to medium high and cook farro until it is tender (usually about 25 to 30 minutes); strain the farro. Set aside on a paper towel to leach as much of the water as you can off of the farro.
In a small pot, heat the peanut oil to 350 degrees F. Add the farro and fry until crisp, about 1 to 1½ minutes. You want them crisp, but with a bite to them.
Arrange the sliced tomatoes on a large platter and season them with the remaining salt. Drizzle 1½ tablespoons of the vinaigrette over the tomatoes.
In a large bowl, combine the cucumber and arugula. Dress with the remaining 1½ tablespoons of vinaigrette and toss well.
For serving, layer the cucumber and arugula in the center of the tomatoes on the platter. Garnish with the crisp farro. Add fresh ground black pepper, to taste.