PreparationMake the pickled vegetables: Bring vinegar, brown sugar, salt, and 2 cups water to a boil in a large saucepan. Meanwhile, pack vegetables into jars. Pour brine over vegetables. Cover and chill until cool. Make the dressing and assemble: Whisk buttermilk, yogurt, lemon juice, and 3 Tbsp. oil in a large bowl until smooth; season dressing with salt and pepper. Scoop avocados into a small bowl; add 1 Tbsp. oil and lightly mash. Season with salt and pepper. Mash goat cheese with remaining 2 Tbsp. oil in another small bowl until softened and spreadable; season with salt and pepper. Add lettuce to dressing and toss to coat; season with salt and pepper. Spread avocado mixture over 4 slices of bread. Arrange lettuce over and top with cucumber, sprouts, and some drained pickles. Spread remaining 4 slices of bread with goat cheese and close sandwiches. Do Ahead Pickles can be made 2 weeks ahead. Keep chilled.