For the piquillo pesto: Combine the peppers, Parmesan, walnuts, olive oil, salt, chile paste and garlic in a food processor. Puree until smooth and evenly combined. Set aside.
For the roasted tomatoes: Preheat the oven to 350 degrees F.
Toss the tomatoes with the basil, garlic, olive oil and salt on a rimmed baking sheet. Spread evenly on the baking sheet and roast until softened and beginning to blister, 30 minutes.
For the mushroom tapenade: Heat a large skillet over medium-high heat. Add 2 tablespoons olive oil and the shallots and cook until fragrant, about 1 minute. Add the mushrooms and season with the salt. Cook, stirring often, until the mushrooms have lost all their liquid and are turning golden brown, about 10 minutes. Remove from the heat and season with the basil, oregano, rosemary and lemon zest. Stir in the pine nuts and remaining tablespoon olive oil. Cool slightly. Serve warm or at room temperature.
For the bar: Display all the different types of mozzarella and burrata on platters and plates. Set up a buffet around the cheese options with the 3 homemade toppings, along with the grilled bread and crostini, arugula, basil, olive oil, balsamic vinegar, flaky salt and red pepper flakes. Invite guests to assemble their own bruschetta or salads.