Place warm water in the bowl of a stand mixer fitted with a dough hook OR a large bowl. Dissolve yeast in water and let stand 5 minutes until foamy.
Mix brown sugar and salt into yeast mixture until smooth. Add 2 cups flour to bowl and mix on low speed OR with a wooden spoon if making dough by hand until dough pulls away from sides of bowl, about 2 minutes of mixing (if dough seems a bit wet, add additional flour by the tablespoon until dough pulls away from sides of bowl)
Add softened butter while mixer is running or fold in by hand with a wooden spoon until butter is combined in the dough.
If making dough in stand mixer, knead dough in mixer on low speed 5-7 minutes until dough is stretchy and elastic. If making dough by hand, turn dough onto a lightly floured surface and knead by hand until dough is stretchy and elastic, about 10 minutes of kneading.
Place dough in a greased bowl, turning once to grease top. Cover bowl with plastic wrap and let rise in a warm place at room temperature 1 to 1-1/2 hours until dough is doubled in size.
Line 2 large sheet pans with parchment paper. Punch risen dough down and divide into 10 equal pieces. Roll each portion into a 8-in rope and place 2 inches apart on prepared sheet pans. Cover breadsticks with a kitchen towel and let rise an additional 20-30 minutes until breadsticks are puffy.
Preheat oven to 400F. Brush breadsticks with egg white wash and bake at 400F 10-12 minutes until golden brown. Cool on a wire cooling rack.