Fill a medium saucepan with heavily salted water and bring it to a boil over high heat. Prepare an ice water bath by filling a large bowl halfway with ice and water; set aside.
Remove the fava beans from their pods. Boil the favas in the salted water until the bean inside the outer skin is bright green and firm but not hard, about 1 to 2 minutes. Using a slotted spoon, transfer the favas to the ice water bath to cool. Reserve the fava cooking water. Peel the light green outer skin from each bean to reveal two bright green inner halves; discard the skins. Place the beans in a medium bowl; set aside.
Return the water to a boil over high heat. Add the potatoes, reduce the heat to medium, and simmer until the potatoes are knife tender, about 5 minutes. Drain and place in the bowl with the favas. Add the lemon juice and mustard to the bowl and stir to combine. Season to taste with salt and pepper; set aside.
Heat the oil in a large frying pan over medium heat until shimmering. Add the shallot and sauté until softened, about 3 minutes. Increase the heat to high, add the mushrooms and a pinch of salt, and cook, stirring occasionally, until the mushrooms are tender and brown around the edges, about 3 to 5 minutes. Season with pepper and more salt. Add the reserved favas and potatoes, stir to combine, and cook until heated through and the moisture in the pan has evaporated, about 1 minute.
Remove the pan from the heat and season the mixture with additional salt and pepper. Divide among 4 plates, top each serving with a poached egg, and sprinkle with the chives.