For the onions – Place the onion slices in a large bowl. Add the oil, salt and pepper and toss with your fingers until well coated. When the grill is ready, place the onion slices in a perforated stainless steel pan and grill until golden-brown, tossing with tongs only occasionally. Transfer to a bowl.
Cook’s note: If you don’t have a grill, preheat oven to 475ºF (246ºC). Place the onion slices in a non-stick roasting pan large enough to hold the onion pieces in a single layer. Drizzle with 2 tablespoons olive oil, salt and pepper. Toss well and spread evenly in the pan. Cover tightly with aluminum foil and bake for 15 minutes until onions are tender. Remove foil. Toss the onions again. Return to oven, uncovered, and bake for 10 to 15 additional minutes, until golden-brown. Remove from heat and transfer to a bowl.
To make the vinaigrette – Heat a small heavy-bottomed frying pan over medium-high heat. Add the coriander seeds and cook until they turn a dark color and are fragrant — about 2 minutes — shaking the pan constantly and taking care not to burn the spice. Transfer to a mortar and grind until coarsely powdered. Place the coriander in a small bowl. Add the lime juice, oil, salt and cayenne. Whisk until well blended and set aside.
When ready to serve (do not prepare these ahead of time as the bread will get soggy), cut the bread in four 4”-long pieces. Slice each piece lengthwise in half. You should have 8 tartines. Grill the bread slices cut-side-down until grill marks appear, or place under the broiler until golden-brown. Top the slices with the grilled onions and the chopped avocado. Drizzle with the vinaigrette. Garnish with the basil leaves and serve immediately.