Preheat oven to 350°F. Line an 8x8-inch square baking pan with parchment paper to remove the blondies more easily, or lightly grease and flour or spray with baking spray.
Brown Butter: Melt butter in a small pan over medium heat, swirling gently. Once the foam subsides, watch and cook the butter until the liquid is light golden brown and brown flecks begin to appear and sink to the bottom of the pan. Continue stirring until the butter is a light caramel color. Immediately remove from heat and pour the brown butter into a heat-safe mixing bowl.
Allow browned butter to cool for about 10 minutes, then add coffee and brown sugar to the bowl. Beat with electric mixer until slightly fluffy, or use a large wooden spoon and beat by hand. Add eggs and vanilla, and beat until fluffy.
Sift together blended oats, flour, baking powder, and salt. You can sift directly into the bowl with the brown butter/sugar mixture to save yourself another dirty bowl. Stir dry ingredients into brown butter mixture until just combined. Fold in chopped chocolate and chopped walnuts if using.
Scrape Dirty Blondie batter into prepared pan. Bake 25-30 minutes, until wooden toothpick inserted into center comes out clean. Remove from oven and immediately sprinkle with flaky sea salt. Cool Dirty Blondies in pan for 15-20 minutes before cutting or removing from pan.