Preheat the oven to 350 degrees F.
To make the crumb topping, stir the flour, brown sugar, cinnamon and pinch of salt together in a bowl. Add the cold butter and cut it into the flour mixture until it resembles coarse sand. Place in the refrigerator while you prepare the batter and cream cheese mixture.
To make the coffee cake batter, place the all-purpose flour, salt, cinnamon, ginger, nutmeg, baking powder, and baking soda in a medium mixing bowl. Stir together until fully combined. Set aside.
In a separate large mixing bowl, add the eggs and sugar. Beat together using a hand mixer for about 30 seconds. Add in the Greek yogurt, milk, vanilla extract, and oil. Beat together until fully combined. Fold in the grated carrots.
Add the dry ingredients into the wet. Stir until just combined.
To make the cream cheese center, add the cream cheese and sugar into a third mixing bowl (you can re-use the mixing bowl using for dry ingredients to save on dishes). Beat together using a hand mixer until the cream cheese is smooth. Add the egg yolk and vanilla extract. Beat together until smooth and all the ingredients are combined.
Pour 3/4 of the batter into a greased 8-inch springform pan or 8-inch baking pan. Spread out the batter evenly. Carefully place dollops of the cream cheese mixture over the batter. Using an icing spatula or knife, gently spread out the cream cheese. Top with remaining 1/4 of the batter, and spread out evenly.
Sprinkle the crumb topping over the batter evenly.
Bake for 45-55 minutes or until the batter is set. A toothpick will come out clean once done baking.
Let cool and enjoy!