In a small jar with a tight-fitting lid, combine garlic paste, pepper, balsamic vinegar, and olive oil. Cap jar and shake vigorously until dressing is emulsified.
With a small, sharp knife, cut ends off grapefruit and oranges. Working with 1 fruit at a time, set it cut side down on a cutting board. Following curve of fruit with knife, slice off peel and white pith. Cut between inner membranes and fruit to release segments; put them in a large bowl. Discard membranes and peel. Pour off juices (or just pour into a glass and drink) to keep salad from getting soggy.
Thinly slice dates and add to bowl along with radicchio, 1/2 cup parmesan shavings, and 1/4 cup almonds. Pour dressing over salad; toss thoroughly to coat. Turn salad onto a large platter or into a bowl and top with remaining 1/2 cup parmesan and 1/2 cup almonds.
Note: Nutritional analysis is per 2/3-cup serving.