Preheat oven to 450 degrees. Place chicken breasts on a rimmed baking sheet. Drizzle
with 2 teaspoons olive oil and season with salt, pepper, and 1 teaspoon thyme. Roast
until cooked through, 35 minutes. When cool enough to handle, shred chicken
into pieces. In a small bowl, stir together lemon juice, shallot, garlic, 2 tablespoons oil, and remaining 1 teaspoon thyme; season dressing with salt and pepper.
Meanwhile, in a medium pot, bring potatoes to a boil in salted water and cook until potatoes are tender when pierced with a knife, 5 to 7 minutes; drain well.
In a large nonstick skillet, heat 1 1/2 teaspoons oil over medium-high. Add half the potatoes in one layer. Cook, undisturbed, until potatoes are golden brown and crisp, then flip and cook
other side, 5 minutes total. Drain on paper towels. Repeat with remaining oil and remaining potatoes.
In a large bowl, toss together chicken,
potatoes, dressing, and spinach.