Preheat oven to 375 degrees.
On a parchment lined sheet pan, drizzle eggplant halves with olive oil on both sides and arrange flesh side down. Add foil packet of beets to the sheet pan as well.
Bake for 1 hour, rotating 180 degrees midway through cooking, until eggplant is very soft and caramelized and beets are tender.
When cool enough to handle, roughly dice beets and add to a food processor.
Carefully scrape out the innards of the eggplant and add to the food processor.
Add all remaining ingredients to the food processor except for edible flowers and pita chips.
Process until very smooth. Add dip to a bowl and chill completely. The flavor dramatically changes between warm and cold, so make sure to serve cold for optimal flavors.
Serve chilled with a large drizzle of good olive oil, a sprinkling of sumac, edible flowers and pita chips on the side for dipping.