Place walnuts in a 9-inch square or round pan and bake in a 350° regular or convection oven, shaking pan once, until golden brown under skins, about 10 minutes.
Meanwhile, cut peel off melon; scoop out and discard seeds. Cut melon into 1/2-inch-thick crescents.
On a large platter or on dinner or salad plates, arrange watercress, melon slices, and berries.
Shortly before serving, drizzle about half the agrodolce dressing over fruit on platter or 1 to 2 tablespoons over each serving. Cut chèvre into slices or break into chunks and distribute over salad. Sprinkle with toasted walnuts and pepper to taste. Offer remaining dressing to add to taste.