Preheat oven to 350 degrees F. Grease pans, line with parchment, grease parchment.
Sift matcha powder through a fine-mesh sieve. Whisk together flour, cornstarch, sifted matcha, baking powder, and salt. Set aside.
Beat butter for 2 minutes until smooth. Add oil and beat 1 minute. Add sugar and beat 4 minutes until pale and very fluffy.
Add eggs one at a time, scraping bowl between each. Add vanilla. Mix until combined.
Add sour cream, mix on low until just combined.
Add flour mixture in 3 additions, alternating with milk in 2 additions. Mix on low, stop when just incorporated.
Divide batter evenly between pans. Bake 28 to 32 minutes until toothpick shows moist crumbs.
Cool in pans 10 minutes. Cool completely on wire rack.
For frosting: melt white chocolate gently. Cool to room temperature. Beat cream cheese and butter until smooth, 2 to 3 minutes. Add cooled white chocolate. Add powdered sugar cup by cup. Add vanilla and salt. Beat on medium 1 to 2 minutes.
Level layers. Spread frosting between layers. Crumb coat, refrigerate 20 minutes, apply final frosting layer.
Dust with sifted matcha before serving.