If you cannot find burrata, fresh mozzarella
is also fine to use.Prepare a gas grill or grill pan, then make romesco sauce:
In a food processor, chop almonds and hazelnuts. Add peppers, garlic, and olive oil; process until combined. Mix in paprika, vinegar, and salt. Set aside.
Lightly coat figs with additional olive oil and grill about 1 minute per side; transfer to a cutting board. Brush bread with olive oil and grill, flipping once, 2 to 3 minutes per side.
Cut figs in half (if desired, leave some whole). Spread a thin layer of romesco on bread. Arrange figs and bread on a platter with burrata and arugula, laying some arugula on top of bread as a garnish.