In a medium (10-inch/25cm) skillet that has a cover, heat 1 tablespoon of the olive oil over medium heat with the onion slices, and three of the whole cloves of garlic. Cook, stirring occasionally, until the onions and garlic soften and start to color, about 5 minutes. Reduce the heat to low and continue cooking, stirring infrequently, until caramelized, 25 minutes.
Add the drained beans, tomatoes, thyme, 1 tablespoon olive oil, and 1 cup of water. Increase the heat until the mixture starts to simmer. Use a fork to mash about 1/8th of the beans against the sides of the pan, which will thicken the mixture nicely as it cooks. Continue to simmer until the mixture thickens, about 5 minutes. When done, the mixture should resemble a loose, soupy stew. Turn off heat, remove the thyme sprigs, and mash the garlic cloves with a fork so they meld with the bean mixture. Cover to keep warm, and set aside.
Place the duck confit thigh is a small-to-medium skillet (preferably not cast iron) skin side down. Add a tablespoon of water to the pan and turn the heat to medium-low.
Cover and cook until the skin is soft enough to remove, about 3-4 minutes. Remove the skin from the leg in one whole piece, if possible. (I used a paper towel to grab the tip of the leg and carefully coax the skin off with a pair of tongs. It's okay if it comes off in several pieces but do your best to get large ones.) If the skin pieces are sticking and/or the duck to the pan, slide a spatula under them to scrape and lift them away from the bottom of the pan. It's okay if everything doesn't look picture perfect - duck confit is the ultimate "falling off the bone" French dish - and the duck and skin will eventually be torn over the toasts. Preheat the broiler if planning to use that to toast the bread.
With the lid off, continue to cook the duck thigh, with the skin in the pan next to it, turning both as they cook, until the thigh is starting to get crisp and the skin is close to crisp. The cooking of the duck and skin will take a total of 10 minutes, although the skin may take a bit longer. Remove the duck to a plate and place the skin on a paper towel to drain and crisp up. During the last few minutes while the duck is cooking, turn on the broiler and toast the bread on a baking sheet on both sides until nicely browned, but not hard or overly crisp, or toast the bread in a toaster. When done, rub the remaining garlic clove over the slices of warm bread.
Top each slice of warm bread with some of the bean mixture, about 1/2 cup each. Top with shredded duck confit meat and crumbled crisp duck skin. Drizzle each toast with a little olive oil and garnish with chopped parsley.