Roll out puff pastry on a floured surface about 2” on each side and cut each sheet into 12 equal pieces. Place each piece into the cups of a 24-cup muffin tin and set aside.
In a sauté pan over medium heat, melt 2 tbsp butter. Add leeks, thyme, salt & pepper, stirring to prevent browning. After about 3-4 minutes, add shallots. Cook until leeks shrink to about half their size and shallots are translucent, about 5-10 minutes. Remove from heat.
Evenly distribute the leek mixture into the puff pastry cups, about 1 tsp each. Add a little chunk of goat cheese and fold the dough in. Melt remaining 1 tbsp butter and brush over the tops of the puffs. Bake at 400° for about 20 minutes or until pastry is golden brown. Serve warm.