Cook the quinoa. Combine the quinoa, water, and 1/4 teaspoon salt in a small pot. Bring to a boil. Reduce to a simmer, cover the pot, and cook for about 15 minutes, until the quinoa is tender and the water has absorbed.
Remove the lid, fluff with a fork, and let sit for a few minutes to absorb any remaining water. Add the goat cheese and stir until completely melted. Season with salt to taste. Let sit for at least 10 minutes to cool and set a bit. Now form into 8 patties, each about 2 inches in diameter. (At this point, you can wrap and refrigerate the patties, then fry them within a couple days.)
Set a preferably nonstick skillet over medium heat. Add the olive oil, adding more as needed to create a circa 1/8-inch thick depth. When the oil is shimmery and hot, add the quinoa patties. (If they can all fit in one pan, great. If not, no big deal—just do batches.) Pan-fry for about 4 minutes until the crust is deeply golden brown. Gingerly flip and fry for another 3 minutes or so. Transfer to a paper towel-lined plate to drain.
Meanwhile, add the arugula to a big bowl. Dress with the olive oil, lemon juice, and salt. Toss and adjust each ingredient to taste. Serve the salad with the hot quinoa patties on top.
Emma is the food editor at Food52. Before this, she worked a lot of odd jobs, all at the same time. Think: stir-frying noodles "on the fly," baking dozens of pastries at 3 a.m., reviewing restaurants, and writing articles about everything from how to use leftover mashed potatoes to the history of pies in North Carolina. Now she lives in Maplewood, New Jersey with her husband and their cat, Butter. Stay tuned every Tuesday for Emma's cooking column, Big Little Recipes, all about big flavor and little ingredient lists. And see what she's up to on Instagram at @emmalaperruque.