Preheat oven to 350°F. Drizzle beets with 1 tablespoon oil and season with salt and pepper. Wrap in foil and place on a sheet tray. Bake until beets are fork tender, 40-55 minutes, depending on the size of your beets. Let cool completely.
Make the dressing. In a small bowl, mix shallot, balsamic vinegar, maple syrup, and Dijon. Stream in remaining 4 tablespoons of oil, whisking constantly until emulsified. Season to taste with salt and pepper.
Once beets are cool enough to handle, wear gloves and use a paper towel to rub the skins off. Discard the skins and slices the beets into ½ inch thick slices.
Assemble the salad. In a large bowl, toss arugula, cooked beets, walnuts, feta, and dressing. Season to taste with salt and pepper, top with fresh basil, and serve immediately.