Preheat oven to 400° F.

Cut off the pointy ends of the garlic heads, drizzle with a little olive oil, and wrap each head in baking foil.  Roast for 30-35 minutes, until the garlic cloves are golden and soft.  Let cool.

Squeeze the garlic cloves into a small food processor or blender.  Add the juice of the lemon, a smashed fresh garlic clove, the olive oil and salt and pepper.  Blend until you get a creamy sauce.  Add more oil and lemon juice if the dressing is too thick.  Taste and adjust the salt if needed.

Drizzle about 1-2 tsp of water over the walnuts, sprinkle with salt, and roast at 350° F for 10-15 minutes.  Check on them so they don't burn.  Let cool.

Bring water to a boil.  Lower the eggs in the water, and boil for 2 minutes and 15 seconds for a very runny yolk.  Remove and plunge into an ice bath.

Preheat oven to 450° F.  Bring a large pot of salted water to a boil.

Clean and cut the Brussels sprouts in half.  Add the sprouts to the boiling water, and cook for 4-5 minutes. Drain and let them dry and cool. Drizzle with olive oil. (you can make this ahead of time and store until ready to roast).

Spread the Parmesan cheese on a plate, and dip each Brussels sprout half into the cheese, and place cut side down on a baking sheet, drizzled with olive oil.  Sprinkle liberally with salt and pepper, and more olive oil.

Roast the Brussels sprouts for 20 minutes, or until soft and crispy.

Cut the lemon into thin slices, drizzle with a little oil and arrange on a tray. Put in the oven, to roast together with the sprouts, but take out about 5 min earlier.  

Mix the Brussels sprouts with a few tablespoons of the dressing, and arrange on a platter.  Add the chopped avocado, the halved quail eggs, the toasted walnuts, and roasted lemon slices.  Serve warm.