Combine the tuna, cucumbers, carrots, onions, oil, parsley, salt, and pepper in a small mixing bowl. Set aside in the refrigerator until needed.
Make the vinaigrette. Stir together the soy sauce, balsamic vinegar, and lemon juice in a medium mixing bowl. Slowly drizzle in the avocado oil, stirring constantly with a whisk. Season with a pinch each of salt and pepper, or more or less as needed.
Place a 3-inch wide by 1-inch tall oiled ring mold in the center of a shallow bowl or plate. Fill with 1/3 cup of the tuna mixture and smooth the top. Slowly and carefully remove the ring to leave a perfect disk of tuna tartare. Garnish the sides of the tuna with th radish slices, slightly overlapping the pieces. Reserve the tartare in the refrigerator until needed.
Poach the quail eggs. Fill a medium skillet halfway with water, stir in ½ teaspoon of salt and 1 tablespoon of vinegar, and bring to a boil over medium-high heat. Reduce the heat to medium-low for a slow simmer. Crack each egg first into a tablespoon measure or small cup, then transfer gently into the water, cooking in batches of 3 or 4 eggs at a time. Poach for 45 seconds for loose yolks, or up to 2 minutes for firmer yolks. Lift the eggs from the water with a slotted spoon and drain over the skillet before transferring to the center of a disk of the tuna tartare. Repeat with the remaining eggs. Sprinkle each with a small pinch of black pepper.
Place a flower on the top of the tartare, or sprinkle a teaspoon of chopped chives over it. Drizzle two teaspoons of the dressing around the tartare and serve.