Transfer to large rimmed baking sheet. Using fingertips, separate skin from chicken over breasts and thighs. Insert 1 thyme sprig between skin and flesh over each breast and each thigh. Repeat with second chicken.
Mix 2 tablespoons coarse kosher salt and 2 teaspoons paprika in small bowl.
Sprinkle all over chickens.
Transfer uncovered chickens, skin side up, to refrigerator; chill at least 8 hours or overnight.
Preheat oven to 400°F.
Place potatoes in 11x7x2-inch baking dish.
Drizzle with 2 tablespoons olive oil and sprinkle with 1/2 teaspoon paprika.
Sprinkle with salt and pepper; toss.
Drizzle chicken with 2 teaspoons oil.
Place chicken and potatoes in oven; roast until thermometer inserted into thickest part of thigh registers 175°F, about 1 hour.
Let chickens rest 10 minutes.
Meanwhile, toss potatoes with 1 tablespoon chopped thyme. Return to oven and roast until tender and skins wrinkle slightly, about 5 minutes. Coarsely smash.
Cut chickens into breast, wing, thigh, and leg pieces.