1. Put the onions in a large skillet over medium heat. Cover and cook, stirring infrequently, until the onions are dry and almost sticking to the pan, about 20 minutes. Stir in 2 tablespoons of the oil and a large pinch of salt and turn the heat down to medium-low. Cook, stirring occasionally and adding just enough additional oil to keep them from sticking without getting greasy. The onions are ready when they're dark, sweet and jammy, 40 minutes to 1 hour later.
2. Sprinkle with black pepper, stir in the thyme and remove the onions from the heat. When they're cool, fold them into the yogurt and stir in the lemon juice. Taste and adjust the seasoning, then transfer the mixture to a serving bowl. (At this point, you can cover the dip with plastic wrap and refrigerate for up to 2 days.)
3. Wipe or wash out the skillet and put it over medium-high heat. When it's hot, add the remaining 4 tablespoons oil. A few seconds later, add half the leeks, turn the heat up to high and sprinkle with salt and pepper. Use a spatula to turn the leeks over as they cook. Be careful: they will go from not-browned to burnt pretty quickly, and you want to catch them in between those stages, when they're browned and crisp. Transfer the leeks to paper towels to drain and repeat with the remaining leeks, adding more oil to the pan if necessary to keep them from sticking.
4. Garnish the onion dip with the crisp leeks and serve immediately with crudités or crackers.