Place an oven rack on an upper middle shelf and preheat to 375°F. Line a baking sheet with parchment or foil.
Carefully remove the caps from each mushroom, then use a damp cloth to wipe away any dirt from the caps. Discard the stems or save for another use. Season the inside of the mushroom caps lightly with salt and pepper.
Heat a skillet over medium-low heat, then add a splash of water. Add the spinach and cover until steam begins to form, about 1-2 minutes. Remove the lid and add the garlic and shallots. Cook, stirring, until the spinach is wilted, the water is evaporated, and the garlic and shallot are fragrant, 3-5 minutes.
Transfer the spinach mixture to a bowl and cool for 5 minutes or until just slightly warm, then stir in the cheese until evenly combined. Spoon the mixture into the mushrooms, gently pressing down to compact the filling.
In a small microwave-safe bowl, melt the butter for 30 seconds, then stir in the breadcrumbs.
Top each mushroom with a sprinkle of breadcrumbs.
Bake for 10, or until the mushrooms have softened and the Boursin is melted. Turn on the broiler to brown the tops, 1-2 minutes.
Top with chives and serve immediately
Notes
Mushrooms can be stuffed in advance and chilled until ready to bake.
Panko breadcrumbs can be substituted for the Italian breadcrumbs.