Wrap beets in aluminum foil and bake for 1-1 ½ hours, or until fork-tender. Let cool. When completely cool, use a paper towel to slip off the skin. Dice in ¼ inch cubes. You should have 1 cup.
Combine may, 3 tablespoons reserved orange juice, 1 teaspoon orange zest, honey, garlic, and salt in a food processor and process until smooth. Cut 4 of the orange sections into quarters. Combine aioli, eggs, roasted beets, and oranges and toss to combine. Season to taste with salt and pepper.