Pulse olives, parsley, capers, orange zest, crushed red pepper, and garlic in a food processor until chopped and evenly combined, about 8 pulses. Add oil and 1 tablespoon of the lemon juice; pulse until combined, about 3 pulses. Cover tapenade; refrigerate until ready to use. Prepare the Halloumi Burgers:
Stir together mayonnaise, grated garlic, 1 teaspoon of the salt, and remaining 1 tablespoon lemon juice in a small bowl until smooth. Cover aïoli; refrigerate until ready to use.
Preheat a grill to medium-high (400°F to 450°F). Brush each halloumi piece with oil; sprinkle halloumi evenly with cracked pepper, cumin, and remaining 1 teaspoon salt. Place halloumi on unoiled grates; grill, uncovered, until browned and charred in places, about 3 minutes per side. Remove from grill.
Spread 1 tablespoon aïoli on each bun half. Place 1 roasted red pepper half on each bottom bun half, then top with 1 halloumi piece. Spoon ¼ cup Citrus Tapenade on top of each halloumi piece. Replace top bun halves. Serve. Aïoli and Citrus Tapenade may be stored in airtight containers in refrigerator up to 2 weeks.