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For the hash browns: Heat 1 tablespoon of the oil and 1 tablespoon of the butter in a medium saute pan over medium heat. Add the onions and a pinch of sugar and cook, stirring occasionally, until the onions are golden brown and caramelized, about 30 minutes. Add the thyme and season with salt and pepper.
While the onions are caramelizing, put potatoes in a medium pot of cold, salted water. Bring the water to a boil and cook until nearly tender but a knife inserted into the centers still meets a little resistance, 10 to 12 minutes. Drain through a colander and then drain on paper towels.
Heat the remaining butter and oil on a cast-iron griddle or in a skillet over medium-high heat. Add the potatoes and cook, pressing firmly with a spatula, until the underside is golden brown, about 5 minutes. Add the caramelized onions and a drizzle of the green onion vinaigrette and toss to coat. Transfer the hash to a platter and drizzle with the remaining green onion vinaigrette.