Preheat oven to 350°F (175°C).

Remove stems from mushrooms and finely chop enough to equal 1/3 cup. Discard remaining stems.

Melt butter in a large skillet over medium heat.

Add prosciutto, chives, and chopped mushroom stems to the skillet.

Cook, stirring frequently, until prosciutto is crisp and mushrooms are tender (about 6 minutes).

Reduce heat to low.

Add cream cheese, Dijon mustard, lemon zest, dried tarragon, and ground black pepper to the skillet.

Stir until the mixture is well combined and smooth.

Spoon the cream cheese mixture evenly into the mushroom caps.

Place the stuffed mushroom caps on a rimmed baking sheet.

Bake in the preheated oven for 20 minutes, or until golden brown and heated through.

Serve immediately.