Cut the radicchio in half and remove the core. Separate the leaves while discarding the wilted, outer ones. Roughly tear into pieces and submerge in a bowl of ice water while you prepare the dressing.
Zest one of the blood oranges into a small bowl, about 1 Tbsp. Using a sharp knife, remove the pith and peel of all oranges. Start by trimming off the very top and bottom of the fruit. Next, set the fruit on one end. Carefully cut the peel from the flesh by starting at the top and following the curves down. Slice 1 blood orange and 1 Cara Cara crosswise in 1⁄4-inch-thick rounds. Supreme the remaining 2 oranges by cutting between the white membrane removing each individual segment of fruit. Remove any seeds and set aside.
In the bowl with the zest, add the shallots, dijon, orange juice, vinegar, dates, honey, and olive oil. Mix to combine and season with salt and pepper.
Dry the radicchio in a salad spinner. Arrange the radicchio on a large platter and spread the citrus on top. Spoon about half of the dressing all over and crumble the goat cheese on top. Sprinkle with the pistachios, a pinch of flaky sea salt, and a few turns of pepper. Serve the remaining dressing on the side.
Notes
Notes from the Kitchen
Up to one week before: Prepare the dressing and refrigerate in an airtight container.
The day before: Segment the citrus and store in an airtight container.