Place a pizza stone or baking sheet in oven. Preheat oven to 500°F. (Do not remove pizza stone while oven preheats.)
Heat oil in a large nonstick skillet over medium. Add mushrooms, bok choy, and shallot; cook 5 minutes, stirring occasionally, until tender. Season with salt and pepper. Set aside.
Combine tomato sauce, miso paste, and fish sauce in a bowl; stir with a whisk.
Roll dough into a circle on a large piece of parchment paper; pierce well with a fork. Place dough (on paper) on preheated stone. Spread tomato sauce mixture evenly over dough. Sprinkle cheese evenly over top. Scatter cooked vegetables over cheese. Bake pizza for 7 minutes. Crack eggs over top of pizza. Bake for another 4 to 5 minutes, or until eggs are set.
Top pizza evenly with kimchi, avocado, and green onion.
Notes
If using spinach in place of bok choy, wait to add the greens during the final minute of cooking the mushrooms and shallots.
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