Put the arugula in a large bowl. In a food processor, pulse the vinegar, olives, mustard, and 1 teaspoon of the thyme. Processing continuously, slowly pour in 1/2 cup of the olive oil and 3 Tbs. water to make a loose dressing. Transfer to a liquid measuring cup.
Heat the remaining 2 Tbs. oil in a large (preferably 12-inch) nonstick skillet over medium-high heat for 30 seconds. Add the diced potatoes, sprinkle with 1 tsp. salt and 1/2 tsp. pepper, and cook, stirring occasionally, until they begin to brown all over, 5 to 8 minutes. Reduce the heat to medium and cook until the pieces are just tender, another 4 to 5 minutes. Add the onion and cook, stirring, until it’s soft and lightly browned, 3 to 5 minutes. Stir in the remaining 1 teaspoon thyme. Transfer to a bowl or plate.
Reduce the heat to low and pour the olive vinaigrette into the pan. Cook, stirring with a wooden spoon or spatula until the vinaigrette is warmed through, about 1 minute. Return the vinaigrette to a liquid measuring cup and whisk to recombine.
Toss the arugula with about half of the vinaigrette or enough to lightly coat. Portion the arugula among four (or six) plates. Top with the potato and onion mixture and then the feta and tomatoes. Drizzle with some of the remaining vinaigrette (you may not need it all) and serve.
If you opt out of topping the salad with a chicken breast, consider serving it with a bowl of Lemon Chicken Soup and crusty bread.