In a small bowl, stir together yeast, warm (not hot) water, and light brown sugar. Allow to sit for 5 minutes. The mixture should foam, signaling that the yeast is active. Then, stir in rum, eggs, melted butter, both extracts, vanilla extract, and salt.
Pour mixture into a standing mixer fitted with a dough hook. Turn the mixer to low speed and slowly add the flour 1 cup at a time. Let the dough knead for 10 minutes. Dough should be soft and smooth, like skin.
Coat a large bowl with oil and place the dough into the bowl. Cover with plastic wrap and set in a warm area to proof for 1 hour.
Once dough has proofed, lightly grease a bundt pan with butter or oil. Sprinkle bottom with 1/3 of chopped pecans. Set pan aside.
In a small bowl, mix brown sugar and cinnamon together. In a separate bowl, melt butter.
Lightly knead the dough in the bowl, then begin to rip the dough into 2 inch balls (in the end you should have roughly 40 balls of dough). Roll each ball between your hands so it’s circular. Dip each ball into melted butter, then into brown sugar mixture, then place in the bundt pan. Once half of your dough is in the pan, sprinkle another 1/3 of the pecans over the dough. Continue filling pan up with remaining dough, then top with remaining pecans.
Once all dough is coated and placed into the pan, cover pan with towel and allow to proof a second time for 1 hour.
Bake the bread
Preheat oven to 350°F. Bake bundt for 50 minutes, or until the bread is medium golden brown on top.
Allow dough to cool for 10 minutes before inverting it over a serving tray.
Make the glaze
To make the glaze, whisk all ingredients in a microwave-safe bowl. Microwave for 15 seconds, then drizzle over bread! Definitely serve this right out of the oven, as homemade breads tend to go stale pretty fast!