Preheat the oven to 350°. Spread the walnuts in a pie plate and toast them for about 10 minutes, until they are golden. Let cool, then coarsely chop.
In a large bowl, whisk the shallot with the Dijon mustard and sherry vinegar and season generously with salt and pepper. Add the olive oil and whisk until smooth. Add half of the blue cheese and whisk until the dressing is creamy. Add the lettuce, fennel, carrots, cucumber, apple, avocado, basil and walnuts and season with salt and pepper. Toss the salad well to coat with the dressing. Top with the remaining blue cheese and serve right away.