Prep the veggies. Wash the tomatoes and green pepper thoroughly. Chop into medium chunks your blender can handle. Peel the cucumber and garlic.
Initial blend. Add tomatoes, green pepper, cucumber, garlic, sherry vinegar, salt, and half the cold water to the blender. Blend on high 2-3 minutes until fully liquid and uniform.
The 5-minute rest. Turn off the blender and let the mixture sit undisturbed for exactly 5 minutes. This settles the natural fibers and juices, preparing them to bond with the oil.
Stream and emulsify. Turn the blender back on to medium-high. While running, slowly drizzle the olive oil in a thin, steady stream through the lid opening, like making mayonnaise. Watch it turn creamy and slightly orange.
Traditional finish. Right before turning off the blender, add the fresh lemon squeeze.
Strain. Pour through a fine-mesh strainer into a pitcher, pressing the solids with the back of a ladle to extract all the juice. Discard the skins and seeds.
Chill and serve. Refrigerate at least 1-2 hours. Serve ice-cold.