Set a rack in the center of the oven and heat to 350°F. Lightly butter a 8.5×4.5-inch loaf pan and line it with a parchment sling so the long sides overhang for easy removal.
In a medium bowl, whisk together the flour, baking soda, salt, and cinnamon; set aside.
Brown the butter: Melt the butter in a small saucepan over medium heat. Continue cooking, swirling occasionally, until the milk solids turn golden and it smells nutty, 3–5 minutes. Scrape into a large mixing bowl and let cool for about 10 minutes.
Whisk the brown sugar and granulated sugar into the cooled browned butter until glossy and well combined, about 1 minute. Beat in the eggs one at a time until the mixture looks slightly thickened and smooth.
In a separate bowl, stir together the mashed bananas, Greek yogurt, milk, and vanilla. Whisk this banana mixture into the butter-sugar mixture until evenly blended.
Add the dry ingredients to the bowl and fold gently with a spatula just until no dry flour remains. Avoid overmixing so the bread stays tender.
Scrape the batter into the prepared pan and smooth the top. Bake 55–65 minutes, or until a tester inserted in the center comes out clean or with a few moist crumbs. If the top is browning too quickly, tent loosely with foil for the last 10–15 minutes.
Cool in the pan on a rack for 15 minutes, then lift out using the parchment and cool completely before slicing. For the cleanest slices, wait until the loaf is just barely warm.