Heat a grill pan or outdoor grill to medium high (about 375°F to 425°F).Meanwhile, arrange the bread slices in a single layer on a baking sheet. Using 1 tablespoon of the oil, very lightly brush the tops of the bread with the oil. Flip the slices over and brush with the remaining tablespoon of oil. Generously season one side only with salt and pepper.
Heat the olive oil in a large frying pan over medium heat until shimmering.
Add the shallot and cook, stirring occasionally, until softened but not browned, about 2 minutes.
Add the radicchio, season with salt and pepper, and cook, stirring occasionally, until mostly wilted, about 5 minutes.
Add the vinegar and brown sugar and stir to combine. Cook until the vinegar has been absorbed, about 4 minutes. Continue cooking, stirring occasionally, until the mixture is completely wilted and dark in color, about 10 to 12 minutes more.
Remove the pan from the heat. Taste and season with additional salt and pepper as needed; set aside.While the radicchio is cooking, place the goat cheese in a medium bowl. Using a rubber spatula, smash the cheese against the sides of the bowl until it reaches a smooth consistency.To serve, spread an equal amount of the goat cheese over each slice of bread. Divide the radicchio mixture evenly over the goat cheese.
Cut the bruschetta into pieces if desired and serve.
