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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Prep time was over an hour for me. I didn't have pre-cooked quinoa so I tried adding it dry (1/2 cup) to the skillet to let it absorb the vegetable juices, the broth and then the meat juices in the oven. This way it did double-duty so that I didn't need the bread crumbs; and the loaves came together just fine. I have a round, 7-well cast-iron muffin/biscuit pan so I used this to make slightly larger meat loaf "muffins." I used feta instead of goat cheese crumbles as this was what I had, and I didn't have ketchup but the loaves didn't need it. They were moist with lots of flavor; not "light" or "healthy"-tasting even with all the quinoa and vegetables. I served them with Southern Living's "Brown Butter Cauliflower Mash" and steamed green beans. This is a great way to stretch a pound of grass-fed ground beef and I'll be making these again, maybe even doubling next time as Kate suggests.