For the caramelized onions: Heat the oil over medium heat in a large nonstick skillet. Add the onions, vinegar, salt and pepper, stirring occasionally, until the onions are deep golden brown, about 20 minutes.
Fries: Drain the potatoes and dry completely on a clean kitchen towel. In a large heavy-bottomed saucepan, pour enough oil to fill the pan about a third of the way. Heat over medium-high heat until a deep-frying thermometer inserted in the oil reaches 375 degrees F. (If you don't have a thermometer a cube of bread will brown in about 5 minutes.)
While the oil is heating, in a large bowl, whisk together the flour, baking powder, baking soda, salt and 2 1/2 cups water until smooth. Coat the potatoes in the batter. Transfer 8 to 10 potatoes to the oil, allowing any excess batter to drip back into the bowl. Fry until golden and crisp, 5 to 6 minutes. Drain on paper towels. Repeat with the remaining potatoes.
For the sauce: In a small skillet, melt the butter over low heat. Add the thyme and garlic and cook for 2 minutes. Pour the sauce over the fries and gently toss until coated.
Place the fries on a platter and spoon the onions on top. Sprinkle with fontina and serve immediately.