Melt butter in medium saucepan over medium-low heat. Add shallot and thyme and cook, stirring occasionally, until shallot is golden, about 5 minutes. Add flour and cook, stirring frequently, for 2 minutes. Add mushrooms and tomato paste and cook until fragrant, 1 minute. Stir in broth, Madeira, and port. Increase heat to medium-high and bring to boil. Cook until reduced to 2 cups, about 15 minutes. Season with salt and pepper to taste. Strain sauce through fine-mesh strainer set over bowl; discard solids in strainer. Return sauce to saucepan and cover to keep warm. (Madeira sauce can be refrigerated for up to 3 days; warm gently before serving.)