Spray tops of mushrooms with olive oil. Place bottom side up on a large rimmed baking sheet and season with salt and pepper.
Heat 1 tablespoon of olive oil in a large skillet over medium heat. Add shallot and garlic; sauté until softened, 2-3 minutes. Add kale and wine, increase heat to high.
Once wine has cooked off (about 3 minutes), add milk and flour. Return heat to medium and cook until sauce has thickened, 3-4 minutes more. Remove from heat.
Divide kale mixture into mushrooms. Top each with 1/4 cup of cheese. Bake for 10 minutes.
Combine panko and 1 teaspoon of olive oil in a small bowl while mushrooms are baking. Top each mushroom with 1 tablespoon of panko mixture and return to oven for another 5-8 minutes, or until mushrooms are tender and panko is golden brown.