Cut trimmed cauliflower into large pieces. Boil or steam cauliflower until just tender, about 5-6 minutes depending on size of pieces. Drain and rinse with cold water until cool. Allow to drain completely and press dry with a paper towel to absorb any excess moisture.
Using a food processor, pulse together cooked cauliflower, oat flour, eggs, salt and pepper until combined and cauliflower is broken down into small bits. (you want some texture remaining) Stir in parmesan cheese, green onions, and parsley.
Heat 1 tbsp of olive oil in a non stick pan over medium heat. Using approximately 1/4 cup of cauliflower mixture form into patties by scooping into a 3" ring mold set in the bottom of the pan and flatten with a spoon. Remove the ring mold and repeat for remaining cakes. Alternatively, you can drop 1/4 cup of mixture into hot pan and flatten into a 3" disk using the back of a spoon. Cook for 4-5 minutes per side until golden and keep warm uncovered in a 200℉ oven. Repeat with remaining cauliflower mixture to make a total of 8 cakes.
Split avocado in half and remove pit. Scoop out flesh into the bowl of a food processor or blender. Add yogurt, olive oil, lemon juice, salt and pepper. Blend until smooth. With processor running, slowly add water until desired consistency is reached. Taste for seasoning and spoon over eggs.
Using the same pan from cooking the cauliflower cakes, gently warm ham or back bacon while eggs are cooking.
Fill a large shallow pan with 2"-3" of water and add 2 tbsp of white vinegar. Heat to a simmer. Poach 8 eggs by first cracking 1 egg into a small bowl and sliding egg gently into the water. Repeat with remaining eggs. Cook for 4 minutes for soft eggs. Scoop out egg using a slotted spoon and rest onto a paper towel lined plate to absorb excess water.
Top each cauliflower cake with 1 ounce of ham or back bacon. Add a poached egg to each and top with a generous spoonful of avocado sauce. Garnish with chives if desired. Serve immediately.
To make oat flour, grind 1/2 cup oatmeal in a spice grinder until flour like.