Place oven rack in lowest rack position heat oven to 400. Spray 12 muffin cups generously with cooking spray. Melt 4 tbsp of butter in medium skillet over medium heat. Cook onion 3 minutes or just until it begins to soften, stirring occasionally.
Meanwhile, whisk 2 of the eggs, 1 tsp salt, and 1/2 tsp pepper in large bowl until blended. Stir in hash browns until coated. Stir in onion mixture. Spoon into muffin cups (cups will be full). With back of spoon, press into bottom and up sides, making an indentation in the center. (Keep hash browns from spilling over edges to avoid over browning).
Bake on bottom oven rack 40-45 min or until potatoes are tender and sides are brown. Remove from oven; run thin metal spatula around outside edges to loosen. Cool in pan on wire rack 5 minutes. Carefully remove potato baskets from pan; place on foil-lined baking sheet (Potato baskets can be made 4 hours ahead).
Place oven rack in middle position; reduce oven temperature to 300. Whisk remaining 8 eggs, 1/2 tsp salt, 1/4 tsp pepper, tarragon, and chives in large bowl 45 seconds or until frothy and blended.
Melt remaining 2 tbsp butter in large non-stick skillet over medium heat. Pour in eggs; reduce heat to medium-low. Cook slowly 2-3 minutes or until eggs just begin to form curds, stirring constantly with heat-resistant spatula. Stir in cheese. Cook 1-2 minutes or until curds form but eggs are still very creamy. Spoon eggs into potato basktes.
Immediately bake potato baskets for 5 minutes or until hot.