Place the chocolate chips in the refrigerator or freezer when you start, so they don't dissolve entirely when you add them later.
Melt butter in a large, microwave-safe mixing bowl, for about 1 minute or until the butter melts. Stir in the sugar and microwave for another 1-2 minutes, stopping it if it starts to bubble. [See notes for instructions on if it bubbles.] Stir to help dissolve the sugar; it's okay if the mixture separates.
If not using the microwave, heat butter and sugar together on the stove until the sugar dissolves.
Remove butter and sugar from heat. If your mixture and bowl are very hot, wait a few minutes to let cool so you don't scramble the eggs.
Use a wire whisk to mix in cocoa powder and vanilla, stirring until combined. Add eggs one at a time, mixing each one in completely.
Use a rubber spatula to fold in flour, baking powder, and salt, mixing gently until completely mixed in. Stir in chilled chocolate chips.
Scrape batter into a greased or parchment-lined 9x13 inch pan, and bake for 30-35 minutes at 350F. Start checking for doneness at 25 minutes if the sugar bubbled, you're concerned you overmixed, or your oven runs hot.
Use a toothpick to test the brownies for doneness. A toothpick inserted into the brownies will come out with moist crumbs but not batter; test a different spot if you hit a chocolate chip. Brownies will look set on top and may have started to pull away from the edge of the pan. DO NOT OVERBAKE.
Wait at least 1 hour, all I can ever stand, before cutting a square to eat right then; those brownies will be fragile and fall apart. Cool at least 2 hours, or until cool, before cutting squares to remove from the pan and serve.
Store in an airtight container at room temperature. Brownies stay good for 3 days, although the edges may stiffen up. Freeze for longer storage.
Notes
The most important tip for brownies is: don't overbake.Measurements: I weigh my ingredients instead of using the volume measurements listed here. Brownies will turn out more dense if you use too much flour.Chilled chocolate chips: Chilled chocolate partially melts into the brownies, providing fudginess in the brownies while also keeping bites of chocolate chips. Cool the batter for 20 minutes to about room temperature before adding chocolate chips if you don't want them to melt at all.Bubbling sugar: Dissolving and sugar into hot butter helps create a shiny, crackly crust. However, if your sugar bubbles, it becomes a syrup and you lose a little moisture. If the sugar bubbles, begin checking the brownies at 25 minutes, as they may be done sooner. Blooming cocoa: Adding cocoa to warm butter helps bring out the cocoa flavor.Don't overmix: Brownies are supposed to be tender, so do mix gently after adding the flour, stirring only until everything's combined.Adapted from the Fudge Brownies recipe at King Arthur Flour.Calorie count is an estimate only.