Heat two separate non-stick sauté pans to medium-high.
After the artichokes are cleaned and cut in half length-wise, place them in acidulated water* to avoid browning. Drain them well and add them to the hot pan. Allow any excess water to simmer away once the artichokes hit the pan.
Once the water has evaporated, coat the bottom of the pan with extra virgin olive oil. Give the pan a little shake to allow the cut side of the artichokes to be completely exposed to the pan, which will allow for better color.
When the artichokes begin to turn blonde, add the diced onions. Cook for 1-2 minutes until translucent, then add the garlic and chili flakes — careful not to let either brown. Season the contents of the pan liberally with salt, then deglaze with the wine. Reduce to medium heat until the artichokes are tender. Once the artichokes are fully cooked, the wine should have reduced to a glaze. Finish the artichokes with extra virgin olive oil and reserve.
NOTE:
This entire process will happen quickly; it is important to have all your ingredients and tools at the ready. Toward the end of the cooking process, it will be key to control the fire to prevent the onions and garlic from burning, and to allow the wine to cook the artichokes.
In the second sauté pan, once heated to medium-high add extra virgin olive oil. Add the eggs one at a time and fry them in the oil until they are cooked sunny side up with a bit of browning on the bottom. Season with salt and pepper.
Divide the artichokes evenly between four plates, cut side up. Place an egg on top of each plate and finish the plate with a drizzle of extra virgin olive oil, mint leaves, and grated Pecorino Romano.
*Acidulated water is water to which a small amount of vinegar, lemon juice, or lime juice has been added.