Bring 2 cups water to a boil. Pour over the bulgur in a large bowl; cover and set aside until tender, about 20 minutes.
Meanwhile, combine the tahini, 1 tablespoon each olive oil and lemon juice, the garlic and a big pinch each of salt and pepper in a food processor. Add 1/2 cup small ice cubes and process, scraping down the food processor as needed, until smooth and all the ice has melted. Remove 1/2 cup to a small bowl for the dressing.
Add 1/2 cup mint, the peas and cumin to the food processor with the remaining tahini mixture and puree until smooth. Add the chickpeas, panko and a pinch each of salt and pepper; pulse until combined but still a little chunky. Scoop the mixture into 12 balls, about 2 tablespoons each.
Heat 3 tablespoons olive oil in a large nonstick skillet over medium-high heat. Add the falafel and cook, turning, until browned, adding 1 more tablespoon oil when the pan gets dry, 6 to 8 minutes.
Pour off any unabsorbed water from the bulgur. Add the cucumber, remaining 1 tablespoon olive oil and 1 tablespoon lemon juice. Tear the remaining 1/2 cup mint and add to the bulgur; season with salt and pepper and toss. Divide among plates; top with the falafel. Thin the dressing with the remaining 1 tablespoon lemon juice if needed and drizzle on the falafel.