In a Dutch oven or heavy-bottomed pot, heat 1/4 cup of the olive oil over medium-high heat. Add the mushrooms, garlic and onion and season with pepper. Cook until the vegetables are softened and lightly caramelized, about 8 minutes. Season with salt and pepper. Add the chicken stock, bread, vermouth and thyme. Bring to a simmer and simmer for another 20 minutes.
Transfer the soup to a blender, add 2 to 3 teaspoons sherry vinegar and, with the motor running, drizzle in the remaining 1/2 cup olive oil for a creamy texture. Season with salt and pepper and more sherry vinegar as needed to balance the richness. Divide the hot soup among bowls and garnish with Tahini Herb Drizzle, fried shallots and chopped dill.
Tahini Herb Drizzle:
In a medium bowl, whisk together the tahini, olive oil, lemon juice, parsley and honey. Season with salt and pepper. If necessary, thin with a few teaspoons of water to achieve the desired consistency.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.