In a large skillet, heat 2 tablespoons of extra-virgin olive oil over medium heat. Once the oil shimmers, add the chopped shallots and sauté for about 3 minutes, stirring frequently until they become translucent.

Next, add the diced pear, 1 tablespoon of maple syrup, and 1/4 teaspoon of kosher salt to the skillet. Stir and allow to cook for another 3 minutes until the pear pieces soften slightly.

Pour in 1/4 cup of white balsamic vinegar and let the chutney simmer for 8 to 10 minutes, stirring occasionally, until it thickens. Add 1 teaspoon of fresh rosemary for flavor, then cool the mixture in a bowl.

Wipe the skillet clean and add the remaining 2 tablespoons of olive oil. Arrange the trimmed Brussels sprouts cut-side down and cook undisturbed for 4 minutes.

Once golden, gently toss them and cook for another 5 to 7 minutes. Add remaining 1/2 teaspoon of kosher salt as they become crisp-tender.

Toss the cooked Brussels sprouts with the prepared white balsamic-pear chutney in a large bowl until evenly coated. Serve immediately.