Blanch fresh peas in a medium pot of salted boiling water until bright green and just tender, 3 to 4 minutes. Transfer to an ice-water bath and let cool completely. Drain. (If using frozen peas, skip this step.)
In a food processor, pulse peas and chickpeas until coarsely chopped. Transfer to a bowl and stir in onion, parsley, egg, and breadcrumbs. Season with salt. Form into 4 patties, each about 3/4 inch thick.
Heat oil in a large nonstick skillet over medium. Cook burgers until golden and crisp, about 4 minutes a side.
Stir together mayonnaise and harissa in a small bowl until combined. Spread on brioche rolls. Dividing evenly, sandwich burgers, baby greens, and cucumber; serve.