Combine the chicken marinade: in a large gallon-size zip-top bag add 1/4 cup olive oil, minced garlic, 1 tablespoon chopped fresh oregano, smoked paprika, crushed red pepper flakes, lemon juice and a pinch of kosher salt and pepper. Add the chicken, toss to coat, seal and refrigerate at least 1 hour or up to 12 hours.
Make the green basil sauce: in a blender or food processor combine 1/2 cup fresh basil, 1/4 cup fresh oregano, 1/3 cup roasted cashews, 2 tablespoons fresh chives, 1/4 cup olive oil and a pinch of salt and pepper; blend until smooth, streaming in about 1/3 cup water as needed to reach a pourable consistency. Taste and adjust seasoning.
Preheat a grill, grill pan, or skillet to medium-high heat. If using, thread marinated chicken onto skewers for easier handling.
Grill the chicken, turning occasionally, until lightly charred and cooked through, about 10 to 12 minutes total depending on thickness. Transfer to a cutting board, let rest 5 minutes, then slice against the grain.
Heat a large skillet over medium and add a drizzle of oil. Add halloumi slices and fry 1–2 minutes per side until lightly golden. Remove and drain briefly on paper towels.
In the same skillet, add the spiralized zucchini/yellow squash noodles and a drizzle of olive oil. Season with salt and pepper and cook 3–5 minutes until just warmed through and slightly softened.
Remove noodles from heat and toss with half of the green basil sauce, halved cherry tomatoes, pitted kalamata olives and the torn basil.
Divide the sauced squash noodles among plates, top with sliced grilled chicken and fried halloumi, and serve with the remaining basil sauce on the side.