Cook quinoa according to package instructions. Meanwhile, bring a medium saucepan with about 2 in. of water and the vinegar to a simmer; keep hot, covered.
Heat 1 tbsp. oil in a frying pan over medium-high heat. Cook onion, carrot, and chard stems, stirring often, until softened, about 10 minutes. Add garlic and mushroom; cook until mushroom softens, adding oil to pan if needed, about 2 minutes. Put chard leaves on top of vegetables, add 2 tbsp. water and the salt, and cook, covered, until leaves wilt, about 2 minutes. Stir in quinoa. Divide mixture between 2 bowls.
Crack an egg into a small dish. Using a slotted spoon, swirl simmering water in a circle, then slowly pour in egg. Cook until white firms and yolk is done the way you like (1 minute for softly set).
Lift egg out of water to top one of the quinoa bowls; repeat with second egg. Drizzle each bowl with 1 tbsp. oil and sprinkle with pepper and chives.