Preheat oven to 400 degrees F. Line fifteen 2 1/2-inch muffin cups with paper bake cups; set aside. In a medium bowl combine flour, baking powder, and salt; set aside. In a large skillet melt 2 tablespoons of the butter over medium heat. Add shallots; cook and stir until tender. Stir in dried apricots and the 2 tablespoons honey; set aside.
In a large bowl combine the remaining 2 tablespoons butter and the goat cheese. Beat with an electric mixer on medium to high speed for 30 seconds. Add brown sugar; beat until combined. Beat in eggs until combined. Alternately add flour mixture and half-and-half to brown sugar mixture, beating after each addition just until combined. Stir in apricot mixture and thyme.
Spoon batter into the prepared muffin cups, filling each about two-thirds full. Bake in the preheated oven for 15 to 17 minutes or until tops spring back when lightly touched. Cool in muffin cups on a wire rack for 5 minutes. Remove from muffin cups. Serve warm. If desired, serve with additional honey.