To make brown butter: Place 10 tablespoons unsalted butter in a medium saucepan over medium heat. Let it melt and continue cooking it. It will make loud chirping noises at first, but suddenly it will stop. That's when it will brown very quickly, so watch it.
When it's a deep amber color with a nutty aroma, remove it from the heat and let the bubbles subside. The browned bits will go to the bottom. Transfer it to a shallow bowl and let the butter cool down a little.
Put 3 eggs, ¼ cup all-purpose flour, ½ cup white sugar, ¼ cup light brown sugar, 3 tablespoons bourbon and 1 teaspoon vanilla extract in a medium bowl.
Add the browned butter, leaving sediments behind in the saucepan. Do this a bit at a time, mixing constantly.
Place 1 ¼ cups pecan halves on the bottom of the pre-baked pie shell. Carefully add the butter mixture. I like to put the liquid in a measuring cup, put the pie in the oven, and carefully add the liquid. That way, I can fill it to the top without moving it.
Bake for 30 minutes or until golden brown. Don't overbake the filling; it should still be slightly jiggly (but not too wobbly) when you take it out, and the pecans should not be burned. It will firm up as it cools. Cool on a wire rack.
Remove from the pan and serve with whipped cream or a scoop of ice cream.
Keep leftovers at room temperature for no more than 2 days, covered. You can then refrigerate them, covered in plastic wrap or in an airtight container to prevent dryness, but I recommend eating this tart the day it's made, or the next one at the most if possible.