Preheat the oven to 400°F (200°C) and grease an 8-inch round cake pan with olive oil. Line the bottom with parchment paper and grease the paper.

In a medium saucepan, heat 1 1/2 tablespoons of olive oil over medium heat.

Add the minced shallot and cook, stirring occasionally, until softened, about 3 minutes.

Add the chopped thyme and rosemary and cook for 1 minute.

Pour in the low-sodium chicken broth and bring to a boil.

Cook over moderately high heat until the broth is reduced to 3/4 cup, about 10 minutes.

Arrange an overlapping layer of very thinly sliced red potatoes in the prepared cake pan.

Season lightly with salt and freshly ground pepper.

Spoon a small amount of the reduced broth on top of the potato layer.

Repeat the layering process with the remaining potatoes and reduced broth, seasoning each layer lightly.

Pour any remaining broth over the top layer.

Cover the pan with a sheet of oiled parchment paper and then a sheet of foil.

Bake the gratin in the center of the preheated oven until the potatoes are very tender, about 1 hour.

Remove the foil and parchment paper and continue to bake until the top is dry, about 10 minutes longer.

Turn the broiler on.

Remove the gratin from the oven and let it rest for 5 minutes.

Invert the gratin onto a heatproof plate.

Carefully remove the parchment round.

Broil the gratin 6 inches from the heat until the surface is lightly browned, about 2 minutes.

Cut into wedges and serve immediately.