The secret to deliciousness in this crowd-pleaser? A caramelized shallot -- it coaxes the sweetness out of the peas. Spoon the yolk onto the bread for an open-faced sandwich.
Heat 2 tablespoons oil in a pan over medium heat. Add shallot and cook, stirring, until caramelized, 3 to 4 minutes. Add peas and 1/2 cup water and cook, covered, until peas are tender but still bright green, 2 to 3 minutes. Mash with a potato masher and add mint and remaining oil. Stir until combined and season with salt and pepper. Set aside.
Place eggs in a small saucepan and cover with water. Bring to a boil and cook 2 minutes. Remove and immediately cut tops using a serrated knife. Season with salt and pepper.
Spread mashed peas on bread and serve with eggs. Garnish with mint.