Remove stems from mushrooms and finely chop stems.

Heat olive oil in a skillet over medium heat; cook and stir chopped mushroom stems, shallot, and garlic until softened, about 5 minutes. Pour wine into the skillet and stir until liquid evaporates, 1 to 2 minutes. Let the mixture cool to room temperature.

Preheat an outdoor grill for medium heat and lightly oil the grate.

Brush olive oil mixture over the top of each mushroom and place them top-side up on a grilling pan. Mix pesto, pine nuts, and cooked mushroom stem mixture in a bowl, then fill each mushroom. Sprinkle Italian cheese blend over the filling.