Make dressing: Stir ingredients together in a small bowl and set aside.
Make salad: Toss fennel and lemon juice in a small bowl. Put arugula and cherries in a large bowl; set aside.
Heat a large nonstick frying pan over medium-high heat. Add halloumi and oregano to dry pan; cook, turning once, until cheese browns (it will exude brine, then dry out), 4 to 6 minutes total. Remove from heat and discard oregano.
Toss arugula and cherries with just enough dressing to coat, then toss with fennel. Divide among plates. Top salads with halloumi. Drizzle dressing over cheese.
*Find at well-stocked supermarkets.