Adjust oven rack to middle position and preheat oven to 425°F (220°C).
In a 12-inch stainless steel straight-sided sauté pan or skillet, or cast iron skillet, heat 2 tablespoons (30ml) olive oil medium-high heat until shimmering.
Add potatoes to skillet, with one cut side down, in a single layer and evenly spaced, and season with salt.
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 browned on bottom side, 2 to 3 minutes longer, adjusting heat as needed if some of the pieces brown too quickly.
Using a thin metal spatula, flip potatoes onto second flat side.
Continue to cook until browned on second side, 4 to 5 minutes.
Stir in garlic, and cook until aromatic, about 1 minute.
Add remaining olive oil, chicken stock (or water), 3 tablespoons (45ml) lemon juice, and oregano.
Bring to a boil, swirling pan occasionally, and season with salt to taste.
Transfer skillet to oven and roast until potatoes are completely tender, offering little to no resistance when poked with a paring knife, and most of the stock has evaporated, 25 to 30 minutes.
Remove from oven, add remaining 1 tablespoon (15ml) lemon juice, and stir to combine.
Season with salt to taste, then serve.
