Heat a grill to medium-high (about 400 degrees F). Make sure your grill is VERY clean or the halloumi may stick.
In a small bowl, stir together 1 1/2 tablespoon of olive oil and oregano.
Slice the Halloumi cheese into 1/3- to 1/2-inch slices.
Brush the marinade over both sides of the halloumi.
Generously oil the grill. Grill the halloumi for 2 to 6 minutes, turning once or twice, until grill marks appear. When lightly pressed the halloumi should give but not be runny (halloumi is a firm, not melty cheese). Place the lemon on the grill with the cut-sides down and let cook until grill marks appear. (To grill the pita, brush it with olive oil, then cook a minute or two on each side, until hot.)
Transfer the grilled halloumi and lemon to a platter. Drizzle the cheese with the remaining 1/2 tablespoon of olive oil. Squeeze the grilled lemon over the top. Sprinkle with thyme. Serve warm with pita, mixed into a salad, or however you please.
TO STORE: Halloumi is best enjoyed immediately after cooking but can be refrigerated for up to 2 days.
TO REHEAT: Rewarm it in a nonstick skillet over medium heat.
TO FREEZE: Can halloumi cheese be frozen? It sure can! Freeze slices of uncooked halloumi cheese in a vacuum-sealed bag or airtight container for up to 6 months. Then you can thaw in small batches and grill on-demand.