Slice the baguette half on the bias into 3/4-inch-thick slices (about 10). Place the slices on a baking sheet. Bake until lightly toasted, 8 to 10 minutes.
Meanwhile, place a saucepot over medium heat. Add the frozen cherries and cook until softened, about 5 minutes. Use the side of a spoon to cut the cherries in half. In a small bowl, combine the cornstarch with 1/4 cup water. Pour the cornstarch mixture into the cherries, stirring to combine. Continue to cook until slightly thickened, 5 to 8 minutes more. Set aside to cool completely.
To prepare the peanuts, place a large skillet over medium-high heat. Add the butter and heat until melted. Spread out the breadcrumbs in the skillet and cook, stirring frequently, until toasted, about 3 minutes. Push the breadcrumbs to the side of the skillet and add the peanuts. Heat for 1 minute just to cook off any brine. Turn off the heat and stir the peanuts into the breadcrumbs.
To assemble, spoon the cherry sauce onto the crostini. Sprinkle the peanut crumble over the cherries. Sprinkle the crostini with the fresh mint and serve immediately.