In a large bowl, with a mixer on medium speed, beat eggs to blend.
Add ½ cup butter, vanilla, sugar, and banana; mix well.
In another bowl, stir together flour, salt, baking soda, and baking powder.
Add dry ingredients to banana mixture, stir to moisten lightly, then add buttermilk and mix well. Stir in pecans.
Scrape batter into a buttered 5×9-inch loaf pan.
Bake in a 325°F oven until bread is well browned and begins to pull away from pan sides.
Let cool in pan on a rack about 10 minutes; run a thin-bladed knife between bread and pan sides.
Invert pan to remove bread; turn loaf rounded side up and let cool on rack.
Serve warm or at room temperature.
If making bread up to 2 days ahead, cool, wrap airtight, and store at room temperature.