Reduce: Combine balsamic vinegar, brown sugar, and black currant seasoning in a small saucepan over medium heat. Stir until the sugar dissolves. Bring to a gentle simmer and cook, stirring occasionally, until the glaze has reduced by about half and coats the back of a spoon, about 12-15 minutes. Remove from heat — it will thicken further as it cools. Set aside to cool completely before using.
Make pesto: Pulse basil, parmesan, pistachios, garlic, and salt in a food processor. Drizzle in olive oil until desired consistency. Taste and adjust seasoning.
Spread the pesto: Spread a generous layer of pistachio pesto on bottom slice of the schiacciata.
Add the arugula: Add a generous handful of arugula on top.
Layer the mortadella: Layer mortadella slices on the bottom half.
Add the burrata or stracciatella: Break open the burrata and spread it over the mortadella letting the creamy interior coat everything, or spoon stracciatella directly over the top.
Drizzle pesto and the glaze: Drizzle more pesto and the black currant balsamic glaze generously over the arugula.
Close and serve: Press the top half of the schiacciata down gently. Serve immediately.