In a medium bowl, whisk together the flour, granulated sugar, brown sugar, and cinnamon. Pour in the melted butter and mix with a fork until small clumps begin to form. The streusel will look soft and slightly wet. Chill uncovered in the refrigerator while you prepare the cake.

Preheat the oven to 350°F and line a 9-inch springform pan (or 9x9-inch square pan) with parchment paper and lightly grease the sides. Set aside.

In a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment, beat the butter, vegetable oil, granulated sugar, and brown sugar on medium-high speed for 3-4 minutes. The mixture should look pale, light, and fluffy.

Add the eggs one at a time, beating well after each addition. Mix in the vanilla extract until combined, then scrape down the sides and bottom of the bowl to ensure everything is evenly incorporated.

Add the sour cream and Irish cream. Mix on medium speed until the batter is smooth and glossy.

In a separate bowl, whisk together the flour, baking powder, baking soda, salt, and nutmeg. Add the dry ingredients to the wet ingredients and gently fold them together with a rubber spatula until just combined. Stop mixing as soon as the last streaks of flour disappear to avoid overmixing.

Add the batter to the prepared pan, spreading it into an even layer with an offset spatula. Break the streusel into small crumbles with a fork, then sprinkle it evenly over the top of the batter.

Beat the cream cheese, powdered sugar, Irish cream, and heavy cream together in a small bowl until smooth.

Drizzle the icing over the completely cooled coffee cake. Slice and enjoy!