Combine water, polenta, herbs, and 3/4 teaspoon salt in a heavy medium saucepan and bring to a boil over medium heat, whisking. Reduce heat to medium-low and cook, stirring constantly with a long-handled wooden spoon, until polenta begins to pull away from side of pan, 15 to 20 minutes. Stir in butter until incorporated, then transfer polenta to baking dish, spreading evenly with a dampened rubber spatula. Chill, uncovered, until set, about 45 minutes.
Preheat broiler. Line a baking sheet with foil or use silicon sheet.
“Unmold” polenta, then cut into 16 (about 4- by 1-inch) sticks. Brush tops with oil and space evenly on baking sheet. Broil about 4 inches from heat until golden, 15 to 20 minutes.