Position rack in the center of the oven and heat to 450 degrees F. In a small bowl combine 2 tbs. of the oil and about two-thirds of the minced garlic. Brush the insides of the mushroom caps with the garlic oil and sprinkle with salt and pepper. Arrange the mushrooms oiled side up on a rimmed baking sheet and roast about 10 minutes.
In a medium bowl, mix the cream cheese, mayonnaise, and 1/2 tsp. of the thyme with the back of a wooden spoon. Stir in the spinach and the artichokes and season to taste with salt and pepper. In another medium bowl, combine the remaining garlic,1 Tbs. oil, and 1 tsp. thyme with the panko crumbs and cheese.
Spoon the artichoke mixture evenly into the mushroom caps and sprinkle with the crumb mixture. Bake until the crumbs are golden-brown and the filling is hot, about 10 minutes. Serve immediately.