In a large bowl, sprinkle yeast over 1 1/2 cups warm (110°) water. Let stand until softened, about 5 minutes. Stir in 1/2 teaspoon salt and 1 tablespoon olive oil. Gradually mix in 3 1/2 cups flour until a soft dough forms.
If using a mixer, beat with a dough hook on high speed until dough no longer feels sticky and pulls cleanly from sides of bowl, 5 to 7 minutes. If dough is still sticky, beat in more flour, 1 tablespoon at a time.
If kneading by hand, scrape dough onto a lightly floured board. Knead until smooth, springy, and no longer sticky, 15 to 20 minutes; add flour as needed to prevent sticking. Place dough in an oiled bowl and turn to coat.
Cover and let rise in a warm place until doubled, 35 to 45 minutes.
Meanwhile, in a 10- to 12-inch frying pan over medium-high heat, stir onions in remaining 1 tablespoon olive oil until very limp, 25 to 30 minutes. If onions start to burn or stick, reduce heat and stir in 1 tablespoon water. Add balsamic vinegar, 1/4 teaspoon pepper, and 1/2 teaspoon salt and stir until liquid is evaporated, 1 to 2 minutes longer.
Scrape dough onto a lightly floured board; press gently to expel air. Divide into four equal pieces. Place pieces on floured board and cover with plastic wrap; let rest 10 minutes. Roll or stretch one piece at a time into a 13- by 7-inch oval about 1/16 inch thick. Place each oval on a cornmeal-dusted 12- by 15-inch rimless baking sheet; stretch dough, if needed, to reshape.
Arrange about 1/2 cup red onions evenly over each oval. Drop ricotta in 1/2-tablespoon portions over onions (about 12 on each oval); arrange figs, cut side up, around cheese. Sprinkle about 2 tablespoons walnuts and 2 tablespoons blue cheese (if desired) evenly over each oval.
Bake in a 450° regular or convection oven until crust is well browned on top and bottom, 15 to 20 minutes (10 to 15 minutes in a convection oven). If baking two sheets in one oven, switch their positions halfway through baking. Slide each flatbread onto a cutting board or plate. Mound arugula leaves equally on top; cut each oval in half lengthwise, then crosswise into eight slices. Add salt and pepper to taste.
Nutritional analysis per appetizer serving.