Preheat oven to 400˚F. Stand each potato cut-side up. Using a 2 inch circle cutter, cut a cylinder from each piece of potato and discard or reserve the scraps.
Transfer potato cylinders to a bowl and toss with 1 tablespoon olive oil, and salt and pepper.
Place a large, oven-proof pan over medium-high heat, add remaining oil and brown potatoes, about 4 to 5 minutes, on each flat-cut side. (If potatoes brown too quickly lower the heat to medium)
Add butter, thyme and garlic to the pan and continue to cook for 2 to 3 minutes, basting potatoes with melted butter.
Pour warm broth into pan and transfer potatoes to oven and bake for 15 minutes.
Remove potatoes from oven and baste well with butter-stock mixture. Return pan to oven and continue to bake potatoes for an additional 20 minutes. (If all the liquid evaporates, add a little more broth to the pan, as needed).
Check the doneness by inserting a knife through the center of one of the potatoes. Potatoes should be completely fork tender. Remove from the oven.
Top with fresh thyme sprigs and sea salt flakes.
