Move an oven rack to the center position and preheat the oven to 375°F (190°C). Lightly grease a 9x9-inch baking dish with nonstick cooking spray or softened butter, coating the corners.
Wash and dry the pears. Peel them, cut in half lengthwise, scoop out the core and seeds, and remove the stem and woody strand. Slice into 1/4-inch thick pieces and place in a large mixing bowl.
Toss the pears with lemon juice, brown sugar, flour, cinnamon, ginger, nutmeg, salt, and vanilla until every slice is coated. Let sit for about 5 minutes while you make the topping.
Pour the pears into the greased baking dish, scraping the bowl. Spread into an even layer with the back of a spoon and gently press down to remove big air gaps.
In a second bowl, stir together the oats, flour, brown sugar, cinnamon, and salt. Cut in the cold cubed butter with a pastry cutter or your fingertips until coarse crumbs with some pea-sized clumps form. Stop once it looks like wet sand with a few larger crumbles.
Scatter the oat mixture evenly over the pears, covering edge to edge. Leave the topping loose and crumbly instead of pressing it down, paying special attention to the corners.
Place the dish on a rimmed baking sheet and bake on the center rack for 35 to 40 minutes, until the topping is deep golden and thick juices bubble around the edges. If browning too fast, loosely tent with foil for the last 10 minutes.
Remove from the oven and set on a wire rack. Let rest for 10 to 15 minutes so the juices thicken, then serve warm in bowls with your favorite toppings.