Begin by removing the seeds from 2 Tbs. of cardamom pods (this will be for both the filling and dough). Grind them with a mortar and pestle, or in a spice grinder, until fairly fine. I personally don't mind a few chunky bits, but to each their own. Set aside 1 tsp. of the ground cardamom for the filling.
In a small saucepan, heat the milk and 1 1/2 tsp. of cardamom until warm. You don't want it to get too hot, or it'll kill the yeast. Around 100-115* F (38-45* C) should do.
In the bowl of a stand mixer with a dough attachment, or a large mixing bowl, mix the cardamom-milk mixture and 1 Tbs. of brown sugar. Sprinkle the yeast over the top, and let it stand for 10 minutes, until foamy.
Add the remaining sugar, salt, egg, and butter to the milk-yeast mixture. Lightly mix to combine (it's ok if it's still lumpy). With the mixer running, begin adding the flour one cup at a time. Start with 3 cups and add more as necessary up to 4 cups total. You want the dough to be smooth and slightly sticky. When you push it with your finger, it should bounce back. Once it reaches this point, knead the dough for about 4 or 5 minutes, until nice and smooth. Cover the bowl with a tea towel, and let the dough double in size, about 40 minutes - 1 hour.
Make the filling while the dough rises. In a small bowl, combine the butter, brown sugar, cinnamon, cardamom, and vanilla extract. Set aside.
Line two cookie sheets with parchment paper, and set aside. Tip the dough onto a clean, lightly floured surface. Roll into a 22'' x 14'' rectangle. Spread the filling evenly, out to the edges. Fold the dough into three sections, like a business envelope. Roll the dough slightly, lengthwise, to lightly seal the dough. Cut the dough in about 1'' sections. You will have about 18-22 strands.
Working one at a time, twist each dough strand. Fold into a light knot, folding the ends under the bottom. (Refer to the video linked above for an example). Place on a baking sheet. Repeat with the remaining strands of dough, placing them about 2 inches apart from each other. Cover the buns with a tea towel, and let rise for about 30-40 minutes. Meanwhile, preheat the oven to 400* F (205* C).
While the buns are rising, mix the water, sugar, vanilla extract, and rose petals in a small saucepan. Bring to a simmer for a couple of minutes. Remove from heat, and let the rose petals infuse for another few minutes. Strain, if desired.
Once the buns have risen, bake for 8-10 minutes or until golden brown.
Remove from oven, and immediately brush with the rose syrup. Sprinkle with pearl sugar, rose petals, and extra cardamom if desired. Store in an airtight container for up to three days.