1/2 cup pearl, hulled, or hull-less barley
1/4 cup fine fresh breadcrumbs
1 tablespoon extra-virgin olive oil, plus more for drizzling
2 cups whole milk
1 cup finely chopped cauliflower
1 small shallot, finely chopped
1/2 cup finely chopped celery hearts, plus 1/4 cup finely chopped leaves
1/4 cup finely chopped fresh flat-leaf parsley, plus 1/4 cup finely chopped fresh flat-leaf parsley stems
1/2 teaspoon finely grated lemon zest
2 tablespoons fresh lemon juice
1 tablespoon Champagne vinegar or white wine vinegar
8 ounces burrata or fresh mozzarella, torn
1 tablespoon crème fraîche
Coarsely ground black pepper
Preheat oven to 350°F. Cook barley in a large pot of boiling salted water until tender, 15–20 minutes for pearl, 35–40 for hulled or hull-less. Drain and spread out on a rimmed baking sheet; let cool.
Meanwhile, toss breadcrumbs with 1 tablespoon oil on another rimmed baking sheet; season with salt. Bake, tossing once, until golden brown, 10–12 minutes; let cool.
Bring milk to a simmer in a medium saucepan over medium-high heat; season with salt. Add cauliflower and cook until just softened, about 3 minutes; drain well. Discard milk.
Toss cauliflower, barley, breadcrumbs, shallot, celery hearts, celery leaves, parsley, parsley stems, lemon zest, lemon juice, and vinegar in a bowl; season salad with salt.
Mix burrata and crème fraîche in a medium bowl; season with salt. Divide burrata mixture among plates, drizzle with oil, and top with barley salad; season with pepper.
DO AHEAD: Barley can be cooked 1 day ahead; cover and chill. Breadcrumbs can be toasted 1 day ahead; store airtight at room temperature.