Blend all the marinade ingredients in a blender, or keep in a bowl and blend with an immersion blender.
Wash the zucchini and yellow squash and dry them. Split them lengthwise into two pieces, then place in a sealable plastic bag with the marinade and let sit at room temperature for 1 hour.
Pre-heat the barbecue or grill to medium heat.
Remove the zucchini and yellow squash from the bag and let them drain, discarding the marinade. Lightly pat with a paper towel, then toss them all in a bowl with the EVOO.
Grill the veggies cut-side down over medium heat for 5 minutes with the barbecue lid closed. Flip over and repeat.
Place the veggies on a platter, then season with the salt, pepper, shaved Pecorino Romano and chopped chives. Finish by drizzling with honey and garnish with the reserved scallion greens.