For the pesto: Preheat the oven to 350 degrees F. Spread the walnuts on a baking sheet and toast, tossing once, until fragrant, 6 to 8 minutes.
Bring a medium pot of salted water to a boil over medium-high heat. Add the broccoli and cook until tender, about 4 minutes. Transfer the cooked broccoli to an ice bath and let cool, about 3 minutes. Remove from the ice bath and drain well. Transfer the broccoli to a food processor. Add the garlic, walnuts, honey, lemon zest, lemon juice and 1/2 teaspoon salt and process until finely chopped. With the machine running, gradually pour in the olive oil. Transfer the pesto to a small bowl and stir in the Parmigiano-Reggiano. Cover and set aside.
For the sandwich: Preheat a grill pan over medium-high heat.
Grill the bread until toasted and golden brown, about 2 minutes per side. Season the chicken cutlets with the pepper flakes and 1/2 teaspoon salt and drizzle with the olive oil. Place the cutlets on the grill pan and cook until golden brown and cooked through, 3 to 4 minutes per side. Squeeze the lemon juice over the cooked chicken while it is still on the grill pan. Remove the chicken to a plate.
To assemble the sandwiches, spread about 1 tablespoon of the pesto on each slice of bread. Place 2 pieces of tomato on 4 of the slices. Top the tomatoes with the chicken cutlets and the remaining slices of bread. Cut each sandwich in half to serve.
Cook's Note: The pesto makes more than enough for the sandwiches. It can be stored in the refrigerator in an airtight container for up to 4 days.
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