Slice the onion into thin rings, place in a small bowl with the juice from ½ of your lime, season with kosher or sea salt, stir and let marinate for about 1 hour.
Meanwhile, slice your eggplant into ¼ inch thick (1/2 cm) slices, then into thin strips.
In a medium shallow pot, add the fresh orange juice, lemon juice, rice vinegar, whole cloves, and cinnamon sticks and bring to a boil. Turn down to a simmer and add the eggplant and let simmer for about 4 minutes. Turn off the heat and let cool in the liquid.
Peel the roasted beets (I usually bake the beets for 1 hour at 400° Fahrenheit /200° Celsius and I wrap them in foil with a little olive oil and kosher salt and maybe a sprig or two of thyme) and dice small.
Dice the apple and the cornichons/pickles into the same size and add all three items into a medium or large bowl.
Fold in the reserved eggplant (remove the cinnamon stick and cloves).
In a separate bowl whisk together the non-dairy yogurt, non-dairy cream (the easiest way to make your own cream is to add equal parts raw cashews and water into a high-speed blender and puree until creamy), Dijon mustard, the juice from ½ a lemon, salt, pepper and a little sugar to taste.
Add the cream mixture to the beet-apple-eggplant, and carefully mix together. Garnish with pickled red onions, capers, and fresh thyme.
Store in an airtight glass container in the fridge, keeps for about 2 weeks.