Peel and trim the onions. Cut them crosswise into 1/2-inch-thick slices. Insert a toothpick horizontally halfway into each slice to hold the onion rings together.

Prepare a medium-low charcoal fire or heat a gas grill on medium low for 10 minutes. Brush the grill grate clean and wipe it with a paper towel dipped in oil. Brush both sides of the onion slices with olive oil and grill, covered, turning every 10 minutes, until very soft and well browned on the outside, 35 to 40 minutes total. (A little blackening is OK, but try not to char the onions.) Stack the onions on a large sheet of foil and wrap loosely. Set aside while you make the glaze.

Pour the balsamic vinegar into a small saucepan and add the thyme leaves, pepper, and salt. Boil, uncovered, over medium-high heat until the vinegar has reduced to about 1/4 cup and has a syrupy texture, 8 to 10 minutes. Let cool briefly and season to taste with salt, if necessary.

Transfer the onions to a dish. Remove the toothpicks and any extremely charred layers, if necessary. If the balsamic glaze has thickened, reheat it gently until pourable. Pour the glaze over the onions and brush to distribute it evenly. Serve warm or at room temperature.