Heat olive oil, ground cardamom, and crushed red pepper in a large skillet over medium-high. Add trimmed and halved Brussels sprouts, cut sides down. Cook until browned, about 7 minutes. Add pomegranate juice and white balsamic vinegar. Cook, tossing often, until sauce is syrupy, another 5 to 6 minutes. Top evenly with crumbled feta cheese (about 1/4 cup) and chopped toasted pecans.