Halve nectarines along the natural seam and remove the pits. Choose fruit that is ripe but still firm.
Stir together olive oil, honey, vanilla extract, and cinnamon in a small bowl to make the basting mixture.
Preheat grill or cast iron grill pan to medium-high heat and oil the surface lightly.
Brush the cut side of each nectarine generously with the basting mixture. Place cut-side down on the grill and cook undisturbed for 3 to 4 minutes until grill marks appear and the surface is caramelized. Flip and cook a further 2 to 3 minutes until tender.
While nectarines grill, divide Greek yogurt between four plates and spread into a thick layer with the back of a spoon.
Place two warm nectarine halves cut-side up on each plate of yogurt. Drizzle generously with honey, scatter toasted almonds or pistachios over the top, and garnish with fresh thyme or mint. Serve immediately.