Because of their thin skin, Meyer lemons can be eaten whole. Arranging the halves on the platter alongside the potatoes adds an extra dimension of color and flavor.
Preheat oven to 450 degrees. Arrange chicken on a rimmed baking sheet. Rub with 1 tablespoon oil and season with salt and pepper. Tie legs with twine and transfer to middle rack. Roast, turning once, until golden brown and a thermometer inserted in thigh reads 165 degrees, about 50 minutes. Remove and let stand 15 minutes before serving. Pour pan juices into a small container and skim fat.
While chicken roasts, toss lemons and potatoes with remaining oil and thyme in a medium bowl and season with salt and pepper. Arrange in a single layer on a rimmed baking sheet and transfer to lower rack. Roast, flipping occasionally, until tender and golden brown, about 40 minutes. Serve with chicken and pan juices.