Preheat the oven to 400 degrees F.
Combine 1/2 cup of the blackberries, the shallots, garlic, honey, mustard, and vinegars in a blender and process on high for 10 seconds. Stop and scrape down the sides before continuing. With the machine running, slowly drizzle the oil into the blender until the mixture is slightly thickened and smooth. Adjust the seasonings with salt and freshly ground black pepper, to taste. Strain the dressing through a fine-meshed sieve into a clean container. Set aside until ready to serve, or store in the refrigerator for up to 1 week. Makes about 1 cup.
Melt the butter in an ovenproof medium skillet over medium heat. When the butter is foamy, add the bread and cook until golden and crispy on 1 side, about 2 minutes. Turn the croutons over and remove from the heat. Top each piece of bread with half of a mutton button and place in the oven. Bake until the cheese is melted and lightly browned, about 1 to 2 minutes. Immediately remove the croutons from the skillet.
Quickly toss the salad greens with the remaining cup of blackberries, the walnuts, and about 1/2 cup of the dressing. Adjust the seasonings with salt and pepper, if needed. Arrange salad onto 4 salad plates and top each salad with a cheese crouton. Drizzle the remaining blackberry dressing around the plate and serve immediately.