Preheat the oven to 350ºF. Line a bread pan with parchment paper.
First, cook 2 tablespoons of the lavender on high heat for about a minute until it begins to release scent. Then, turn the heat down to medium-low and add the butter to the pan. Allow to cook for 2 minutes, stirring occasionally. Your butter should begin to brown and small deposits should form on the bottom of the pan. Once the butter is a medium caramel color, remove from heat and strain out the lavender through a fine sieve, catching the butter in a bowl below. Reserve about 1 teaspoon of the strained lavender.
In a medium bowl, combine the flour, baking soda, and salt. In a separate bowl, mix the eggs, yogurt, sugar, infused brown butter, and vanilla. Fold in the mashed banana and reserved teaspoon of cooked lavender. Gradually fold in the flour mixture.
Pour the batter into the bread pan and bake for 1 hour, or until a knife inserted in the center of the bread comes out clean.
While the bread is cooking, make the lavender syrup. In a small saucepan on medium heat, make a simple syrup with 1/3 cup sugar and 1/3 cup water. Once the sugar dissolves completely, add in 1 tablespoon of lavender. Cook for about a minute, until you can smell the lavender. Set aside.
Once the bread finishes cooking, use a small skewer to poke ~20 holes all over the bread. Place the bread on a wire cooling rack and brush the lavender simple syrup over the top of the bread, then over each side. Allow to cool for 15 minutes, then serve!