Heat a large nonstick skillet over medium-high heat. Add oil; swirl to coat. Add sausage; sauté 3 minutes or until browned. Add onions, thyme, and garlic; sauté 5 minutes. Remove from heat.

Bring milk to a simmer in a large saucepan. Reduce heat to medium-low. Stir in polenta and salt; simmer 7 minutes, stirring frequently with a whisk. Remove from heat. Add cheeses; stir until cheeses melt. Place the polenta mixture in a large bowl; let stand 15 minutes, stirring occasionally. Stir in pepper sauce and eggs. Add sausage mixture; stir well to combine. Spread polenta mixture into a 2-quart glass or ceramic broiler-safe baking dish coated with cooking spray. Cool to room temperature. Cover surface with plastic wrap; refrigerate overnight.

Preheat oven to 425°.

Uncover dish. Bake at 425° for 25 minutes or just until set.