Melt butter with oil in a skillet over medium heat.
Add onions, salt, and sugar. Cook, stirring often, until deep golden, 12–15 minutes.
Stir in garlic and thyme for 1 minute. Deglaze with beef broth, scraping up browned bits. Cool completely.
Heat oven to 400°F. Line a baking sheet with parchment.
Combine sausage, Dijon, Worcestershire, breadcrumbs, pepper, and half the cheese. Fold in cooled onions.
Dust the counter with flour. Roll puff pastry into two rectangles, about 10×12 inches each.
Shape sausage filling into two even logs. Place along one long edge of each pastry. Brush edges with milk.
Roll up tightly, seam-side down. Pinch to seal. Chill 10 minutes if dough feels soft.
Cut each log into 6 pieces. Arrange on the sheet. Brush tops with milk.
Sprinkle with sesame seeds and remaining cheese. Slash tops to vent.
Bake until puffed and deeply golden, 20–24 minutes. Rest 5 minutes before serving.