Preheat oven to 350°.
Place 1 cup milk, egg substitute, and goat cheese in a blender; process until smooth. Combine goat cheese mixture, remaining 2 1/4 cups milk, black pepper, thyme, nutmeg, and garlic in a large bowl, stirring with a whisk. Add bread cubes; stir gently to combine. Let stand 10 minutes.
Place half of the bread mixture in a 13 x 9-inch baking dish coated with cooking spray. Arrange Parmesan cheese, ham, and artichoke hearts evenly over the bread mixture. Top with remaining bread mixture, and sprinkle evenly with fontina cheese.
Bake strata at 350° for 40 minutes or until edges are bubbly. Let stand 15 minutes.