Char bell pepper half directly over gas flame or in broiler until skin is blackened. Enclose in paper bag 10 minutes. Peel and seed bell pepper half; place in mini processor. Add 1/2 tablespoon olive oil; puree until smooth. Transfer pepper mixture to small bowl; whisk in 2 tablespoons olive oil, vinegar, capers, shallot, and sugar. Season dressing with salt and black pepper.
DO AHEAD: Can be made 1 day ahead. Cover; chill. Bring to room temperature and rewhisk before using.
Toast baguette slices; brush with olive oil. Top each toast with half of burrata cheese; sprinkle with salt and pepper.
Combine arugula, radicchio, and parsley in medium bowl; toss with half of dressing. Divide salad between plates. Place 1 toast alongside each salad; drizzle with some of remaining dressing.