Step 1Heat the oven to 250 degrees. In a small saucepan, combine the sugar with 1 cup water and bring to a simmer, stirring until the sugar is dissolved. Add the pecans, reduce the heat to a low simmer and poach until tender, about 15 minutes.
Step 2Strain the pecans, discarding the poaching liquid, and spread the nuts in a single layer on a parchment-lined baking pan. Toast until they are crisp, about 30 minutes. Store at room temperature in an airtight container until needed. Makes slightly more than needed for the recipe.
Step 1In a medium bowl, whisk together the vinegar and one-fourth cup plus 2 tablespoons of the olive oil. Season with one-half teaspoon salt, one-fourth teaspoon pepper and the sugar. Adjust seasoning. Set aside. Makes one-half cup vinaigrette.
Step 2Peel and halve the melon lengthwise. Seed the melon, and cut each half lengthwise into three wedges. Cut each wedge lengthwise into one-fourth-inch slices. Divide the melon slices among six plates (arrange the slices so that they overlap slightly).
Step 3Dress the arugula with 4 1/2 teaspoons vinaigrette, or to taste (it should be lightly dressed). Season to taste with salt and pepper, and divide the salad among the six plates, centering each mound of salad over the melon slices.
Step 4Divide the prosciutto into six portions, draping each portion over the salads. Divide the burrata into six portions and mound over each salad. Drizzle a scant one-fourth teaspoon balsamic vinegar and one-half teaspoon olive oil over each portion of burrata. Top each with a pinch of sea salt.
Step 5Scatter the candied pecans evenly over the six portions, and drizzle one-half teaspoon each of the vincotto and chive oil around the salads. Serve immediately.