Preheat the oven to 350°F (175°C). Line a standard 12-cup muffin tin with paper liners (or grease well).
Make the cream cheese filling: In a medium bowl, beat the softened cream cheese and 1/4 cup sugar until smooth. Beat in the egg and 1/2 teaspoon vanilla until fully combined. Set aside.
Make the muffin batter: In a large bowl, whisk together 1 3/4 cups flour, baking soda, salt, pumpkin pie spice, cinnamon, nutmeg, and ginger.
In a second large bowl, whisk the pumpkin puree, oil (or melted butter), eggs, sugar, and vanilla until smooth.
Add the wet ingredients to the dry ingredients and fold just until no dry flour remains. Do not overmix.
Make the crumb topping: In a small bowl, mix 1/3 cup flour, brown sugar, and cinnamon. Cut in the cold butter with a fork or your fingers until the mixture looks like small pebbles.
Assemble: Spoon batter into each muffin cup until about 1/3 full. Add a heaping teaspoon of cream cheese filling to the center. Top with more batter until cups are about 2/3 full.
Gently swirl the cream cheese into the batter with a knife or toothpick (1–2 swirls is enough). Sprinkle crumb topping evenly over the muffins.
Bake for 20–25 minutes, or until a toothpick inserted into the pumpkin part (not the cream cheese center) comes out clean.
Cool in the pan for 5–10 minutes, then transfer to a wire rack to cool completely. Serve warm or at room temperature.