Combine the cream cheese, goat cheese, butter, Parmesan, lemon juice and garlic in a small food processor and process until very smooth and light. Transfer to a medium bowl and stir in the chives, parsley and dill. Season with salt and pepper. Refrigerate until chilled but still spreadable, about 30 minutes.
Preheat the oven to 350 degrees F.
Spread the bread on a baking sheet and bake, flipping the pieces once, until lightly toasted but not colored, about 10 minutes. Let cool.
To serve, spread the cheese mixture on the toasts. Top with the sliced cucumber and radishes and garnish with dill sprigs.