Wash hands with soap and water.
Gently rub sorrel under cold running water. Tear into bite-sized pieces and place in a food processor.
Add the remaining pesto ingredients and run 5 to 10 minutes or until combined into a paste, adding more oil, as needed.
While the food processor is running, bring 6 cups of water to boil over high heat.
In a separate pan, bring 4 cups of water to a boil over high heat.
Turn off the food processor and taste pesto. Season with salt and pepper as needed, processing well after each addition. Cover and transfer to the refrigerator.
Heat the olive oil in a skillet over medium-high heat. Add the zucchini and red bell pepper. Sauté for 2-3 minutes.
Add cherry tomatoes and black beans. Cook until the vegetables are fork tender and slightly browned.
Once the large pot of water reaches a boil, add buckwheat soba noodles. Stir occasionally, cooking for 8-10 minutes or according to package directions.
Drain and rinse the buckwheat noodles, rubbing them between your hands. Change the water up to three times, as needed, until mostly clear.
To the small pot of boiling water, add both eggs in shell, cover and reduce heat to medium-high.
Wash hands with soap and water after handling raw eggs.
Cook eggs for 6 minutes to an internal temperature of 145 °F on food thermometer.
Transfer eggs to a bowl of ice water and peel.
Assemble by dividing the soba noodles between two bowls.
Top each bowl with half of the vegetable mixture, 1 Tablespoon pesto, an egg, and garnish with thinly sliced green onions.