Brown the butter: Melt butter in a light-colored saucepan over medium heat, stirring, until golden with toasty milk solids at the bottom, 5–7 minutes. Scrape into a large mixing bowl, including browned bits, and let cool 10 minutes until just warm.
While the butter cools, heat the oven to 360°F (182°C) and line two light-colored baking sheets with parchment paper.
Whisk the dry ingredients together in a medium bowl: flour, baking soda, salt, and espresso powder (if using).
Whisk the sugars into the warm brown butter until glossy and slightly thick, about 1 minute. Whisk in the eggs one at a time until smooth, then stir in vanilla and molasses.
Switch to a spatula and fold the dry mixture into the wet just until a few flour streaks remain. Add the oats and chocolate chips and fold until no dry spots remain. The dough will be soft.
Rest the dough at room temperature for 15–20 minutes to allow the oats to hydrate. For extra-thick cookies, scoop dough onto a tray and chill the scoops for 10 minutes before baking.
Scoop 1.5-tablespoon portions (a #40 scoop) onto prepared sheets, spacing about 2 inches apart. Press a few extra chocolate chips on top of each mound.
Bake 11–13 minutes, rotating the pan halfway, until edges are set and lightly browned while centers still look soft. For crinkly edges, tap the pan once on the rack right after baking.
Cool on the sheet for 5 minutes, then transfer cookies to a rack to cool completely.