Prepare a medium-hot grill fire.
Trim off 1/2 inch from the top and bottom of the eggplant. Cut the eggplant lengthwise into 1/2-inch-thick slices. Cut the onion crosswise into 1/2-inch-thick slices. To keep the rings together, insert toothpicks or small skewers horizontally through the rings of each slice.
In a small saucepan over medium-high heat, bring the cream, garlic, and thyme to a boil. Reduce the heat to medium and simmer gently until the cream has reduced by half, 3 to 5 minutes. Remove from the heat.
When the grill is ready, brush both sides of the eggplant and onion slices with the 1/4 cup olive oil. Arrange the vegetables on the grill and cook with the lid closed until the first sides have started to soften and get grill marks, 2 to 3 minutes for the eggplant and 5 to 6 minutes for the onion. Flip and cook the second sides with the lid closed until the eggplant is tender and the onion is nicely charred but not necessarily fully tender, 2 to 3 minutes more for the eggplant and 5 to 6 minutes more for the onion. Transfer the eggplant to a cutting board and sprinkle with 1/4 tsp. kosher salt and 1/4 tsp. pepper. Transfer the onion to a large sheet of foil, stack the slices, wrap tightly, and let rest until residual heat has softened the onion, about 5 minutes.
Remove the skewers from the onion slices. Coarsely chop the eggplant and onion and combine in a medium bowl. Add the cream mixture and the Parmigiano; stir to combine. Taste and add salt and pepper as needed. Brush the bread slices on both sides with oil and grill until lightly golden brown, 30 seconds to 1 minute per side. Top each slice with a generous dollop of the eggplant mixture.