Brown the butter in a saucepan over medium heat, swirling occasionally, until amber and nutty, 5 to 7 minutes. Pour into a bowl and cool completely.

Grind the oats in a food processor until a fine powder, about 30 seconds. Whisk the oat flour with the all-purpose flour, baking soda, salt, and cinnamon.

Beat the cooled brown butter with both sugars on medium speed until fluffy, about 3 minutes.

Beat in the egg, then the yolk, mixing after each. Add the vanilla and lemon juice, and beat until smooth, about 1 minute.

Mix in the dry ingredients on low speed just until combined, then fold in the chocolate chips and walnuts. Cover and refrigerate at least 4 hours, preferably overnight.

Preheat oven to 325°F (160°C). Scoop 2-tablespoon dough balls onto parchment-lined sheets and bake 12 to 14 minutes, until edges are golden and centers look slightly underdone. Cool on the sheet 5 minutes, then transfer to a rack and serve warm.