Bring a large pot of water to a boil over high heat. Gently lower eggs into water and cook for 11 minutes. Meanwhile, fill a large bowl with ice water. Add ¼ cup of the salt and haricot verts to the boiling water during last 4 minutes of cooking. Remove eggs and haricot verts immediately from water and place in ice water. Let cool completely, about 10 minutes. Transfer green beans from ice water to a clean kitchen towel; pat to dry. Slice haricot verts into 1-inch pieces on a bias.

Meanwhile, whisk together Caesar dressing, ½ cup of the Parmigiano-Reggiano, olive oil, lemon juice, 1 teaspoon of the lemon zest, pepper, and remaining ¾ teaspoon salt in a large bowl. Add haricot verts, cannellini beans, chickpeas, tomatoes, and ½ cup of the grated Parmigiano-Reggiano; toss to coat. Season with more salt to taste. Peel and slice eggs lengthwise into quarters and add to chickpea mixture; gently fold to incorporate. Set aside. If making ahead, cover and refrigerate for up to 2 days.

Heat butter in a medium nonstick skillet over medium. Add breadcrumbs and cook, stirring often, until golden brown and toasted, about 2 minutes. Remove from heat and pour into a medium heat-proof bowl. Add remaining 2 tablespoons Parmigiano-Reggiano, remaining ½ teaspoon lemon zest, and salt to taste; stir to combine. Divide chickpea mixture among bowls and drizzle with more Caesar dressing, if desired. Top evenly with breadcrumb mixture and serve.