Heat olive oil in heavy large skillet over medium heat. Add chopped onion, chopped thyme, and crushed red pepper; sprinkle lightly with coarse kosher salt and sauté until onion is tender and golden brown, about 10 minutes.

Add artichokes and minced garlic; stir 1 minute. Stir in diced tomatoes with juice and bring to boil. Reduce heat; cover skillet and simmer 15 minutes to allow flavors to blend.
Meanwhile, cook potatoes in small saucepan of boiling salted water just until tender, about 8 minutes*. Drain. *NOTE: Since the potatoes will continue to cook in the oven, don't overcook them, or they'll become mushy.

Add potatoes and capers to tomato-artichoke sauce; cover and simmer 5 minutes. Season to taste with coarse salt and pepper.

Preheat oven to 375°F.

Pour tomato-artichoke sauce into 13×9x2-inch glass baking dish**. Using back of spoon, make 8 evenly spaced indentations in sauce for holding eggs. Crack 1 egg into each indentation in sauce (some of eggs may run together slightly in spots). Bake until egg whites and yolks are softly set, 12 to 16 minutes.

Carefully remove baking dish from oven; sprinkle grated Parmesan cheese over and serve.

Tomato-artichoke sauce can be made up to 6 hours ahead. Cool slightly, then cover and refrigerate. Rewarm sauce before continuing.

My experience is to consider the eggs ready when they are still a little jiggly. They will continue to cook once you remove them from the oven. The shallower your dish and tomato mixture, the quicker the eggs will cook.

I used a cast iron skillet, so I simply put the entire dish into the oven-no need to pour into a glass dish--OR-- You can put the vegetables into individual ramekins and store overnight and/or fill with an egg and bake according to directions.