Preheat oven to 350°F and line a metal 9×13-inch cake pan with parchment paper and spray lightly with cooking spray. Make sure some of the parchment is over the edge of the pan for easy removal. This may require two sheets, one going in each direction – make sure the one that goes across the shorter distance is under the one that goes along the full length of the pan.
Melt 1 cup unsalted butter in a heavy-bottomed saucepan over low heat. Add 2½ cups granulated sugar and stir until smooth, the butter and sugar have started emulsifying, and no butter is separated from the sugar. Remove from heat and let the mixture cool for 5 minutes, stirring often to release heat.
Add 2 tablespoons canola oil and 1 tablespoon vanilla extract to the butter and sugar and beat them in with a rubber spatula until fully combined and there's no separation between the ingredients (no oil resting on top).
Beat in 5 large eggs one at a time, beating with the spatula or a fork after each one. Mix until fully incorporated.
Sift 1 cup all-purpose flour, ¾ cup + 1 tablespoon Dutch-process cocoa powder, and ¾ teaspoon salt into the butter and sugar mixture, stirring gently with a rubber spatula just until the dry streaks are gone.
Fold most of the 8 ounces dark or milk chocolate into the batter.
Pour the brownie batter into the prepared baking pan, sprinkle the remaining chunks on top, and bake for 25 minutes.
Pull the brownies out of the oven and evenly sprinkle 6 cups mini marshmallows over the top of the brownies, and put them back in the oven to bake for an additional 3 minutes.
Remove the brownies from the oven and allow to cool for a few minutes while the icing is prepared.
Prepare the icing by melting 6 tablespoons salted butter in a saucepan over medium heat.
Sift in 2½ cups powdered sugar and 2 tablespoons unsweetened cocoa powder and whisk until combined, then whisk in ¼ cup whole milk and 2 teaspoons vanilla extract.
Remove from heat and immediately pour over the top of the mini marshmallows, using a rubber spatula to help cover as evenly as possible. The icing will begin to set quickly.
Let the icing set completely before removing the brownies from the pan and slicing into squares for best results. Use a plastic knife for the cleanest cuts. The marshmallows will stick to a metal chef's knife unless it's been sprayed with nonstick spray and wiped down.