Preheat oven to 450°F (230°C). Place stock in a liquid measuring cup or small bowl and sprinkle gelatin over top. Set aside.

Cut off ends of potatoes to give them flat sides, then halve potatoes crosswise. Pat dry with paper towels.

In a cast iron skillet, heat oil or fat over medium-high heat until just beginning to smoke. Season potatoes with salt and pepper and add to skillet, broad side down, in a single layer with space between each piece. Cook, without moving, until potatoes begin to brown around edges, 4 to 5 minutes. Continue to cook, rotating and swirling pan gently to promote even browning and prevent sticking, until potatoes are deeply browned on bottom side, 3 to 4 minutes longer, adjusting heat as needed if some of the pieces brown too quickly.

Flip potatoes. Add butter and cook, swirling constantly, until butter is melted and begins to foam, about 1 minute. Add garlic and thyme, and cook until fragrant, 30 seconds to 1 minute. Add stock and bring to a boil.

Transfer skillet to oven and roast until potatoes are tender, and liquid is reduced to a saucy consistency, 25 to 30 minutes.

Return skillet to stovetop. Transfer potatoes to a serving platter, broad side up, leaving sauce in the skillet; discard garlic. Bring sauce to a simmer over medium heat and cook, swirling and stirring constantly, until sauce is emulsified, 30 seconds to 1 minute. If emulsion appears broken or sauce is too thick, add more stock or water, 1 tablespoon at a time, until you reach the desired consistency. Season to taste with salt and pepper and pour sauce over and around potatoes. Sprinkle with flaky sea salt and serve.