Heat a large skillet over medium-high heat. Add oil; swirl to coat. Add carrot, garlic, thyme, and onion; sauté 3 minutes. Add mushrooms; sauté 6 minutes. Add stock, salt, and pepper; bring to a boil. Cook 3 minutes or until liquid evaporates, stirring frequently. Cool completely.
Place bread in a mini food processor; process until coarse crumbs form. Combine breadcrumbs, quinoa, beef, and eggs. Stir in mushroom mixture and cheese. Shape beef mixture into 8 (4 x 2-inch) free-form loaves; place on a baking sheet lined with parchment paper. Bake at 450° for 15 minutes or until done. Top each loaf with 1 tablespoon ketchup.
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