Prepare an 8x8-inch baking pan by lining it with parchment paper. Move the oven rack to the middle position and heat the oven to 350°F.
Add 7 ounces white chocolate and ½ cup unsalted butter to a medium glass or plastic bowl. Heat on high in 30-second intervals, stirring with a spatula after each interval until melted together. It will likely take a total of 3-4 intervals (1½ to 2 minutes). Set the bowl aside to cool.
In a large bowl, whisk 2 large eggs until they are foamy. Add ⅓ cup granulated sugar and continue whisking until the sugar is dissolved. Add 1 teaspoon vanilla extract and the white chocolate mixture and whisk until thick and pudding-like.
Sprinkle 1 ½ cup all-purpose flour and ½ teaspoon table salt over the mixture and fold it in with a spatula until no lumps remain. Fold in 1 cup chocolate chips.
Spread the batter into the prepared baking pan. Bake for about 30 minutes, until the edges are browned and the center is set. Cool completely in the pan before cutting.
Remove the brownies from the pan and peel off the parchment paper. Cut the brownies into 8-12 squares and serve as is, or for a restaurant-quality dessert, warm the brownies (about 10-15 seconds on high for one brownie should do it), and serve with vanilla ice cream, chocolate sauce, and/or whipped cream.
Notes
If you don't have an 8x8" baking pan, you can also use a 9x9" pan, but the brownies won't be quite as thick. You may need to reduce baking time by a few minutes.I prefer to cut up a plain white chocolate bar, but you can also use white chocolate chips. Use dairy-free white chocolate to make these brownies dairy-free.