Blanch beans in a large pot of boiling, salted water until tender, about 5 minutes. Remove to an ice water bath to stop the cooking. Drain beans and set aside. (She doesn't say to but save the water, you need it for the next step)
In the same pot of boiling water, blanch peas until slightly tender, about 30 seconds to 1 minute. Remove peas to the ice water bath; drain and set aside.
Place beans, peas, tomatoes, eggs, water chestnuts, and green onions in a large bowl and season with salt and pepper.
Add mayonnaise and lemon juice and stir to combine.