In large, heavy bottomed saucepan, add 1/4 cup olive oil and sauté onion, rosemary, bay leaves, 3 thyme sprigs, and garlic cloves for 3 minutes over medium heat. Add leek and continue to sauté until all contents are soft, about 10 minutes. Add beans and water to cover, about 5 to 6 cups. Increase heat to bring to boil, then reduce to low to simmer slowly. Cook for 1 to 2 hours, until most cooking liquid has evaporated and garlic and onions have melted into a silky sauce. Season with salt and pepper and keep warm while preparing mushrooms.

In large skillet, heat remaining 1 Tbsp olive oil on medium-high and, working in batches, if necessary, sauté mushrooms along with 2 remaining sprigs of thyme, seasoning with salt and pepper to taste, until mushrooms are lightly browned and softened, about 7 to 9 minutes. Add vinegar and cook for further 2 to 3 minutes, scraping up any bits from bottom of pan.

To serve, spoon some beans onto plate and top with mushrooms; garnish with picked thyme leaves.