Make the dough: Fill a liquid measuring cup or small bowl with 1/2 cup warm (not hot) water. Add the yeast and sugar and stir with a fork; let sit 15 minutes. The top of the water will bubble and it will smell very yeasty-this is what you want!
Combine the flour and salt in a large bowl. Make a well in the center of the flour and pour in the olive oil and the yeast mixture.
Using a fork, gradually stir the flour into the yeast mixture until mostly combined, then mix with your hands to bring the dough together. Turn out onto a clean surface.
Knead the dough until smooth, dusting with flour as needed. Lightly oil a bowl; add the dough, cover with plastic wrap and set aside in a warm place until doubled in size, about 1 hour.
Prep the toppings: For prosciutto-arugula pizzas, make the chile oil: Bring the chiles and olive oil to a simmer; remove from the heat and let steep 1 hour. (This makes extra oil; store in a sealed container for up to 1 week.)
For puttanesca pizzas, make the topping: Mix the tomatoes, olives, capers and red pepper flakes in a bowl; add salt and drizzle with olive oil.
Preheat the grill: Preheat a grill to medium high (you don't want it to be too hot or the crusts will burn before the cheese melts). Brush the grill to remove any crud.
Roll out the dough: Turn the dough out onto a floured surface and divide into 6 equal pieces. Roll each piece into a ball.
Use a rolling pin to roll out each ball into an irregular oval, dusting with flour as needed.
For cracker-thin crusts, use your hands to gently stretch the dough even more.
Stack the dough between layers of plastic wrap and head to the grill (or wrap and refrigerate overnight).
Grill the crusts: Oil the grill (Anne uses a rolled-up towel and tongs), then carefully add a few pieces of dough.
Grill until the top is bubbly and the bottom is marked, about 2 minutes.
Flip with tongs and grill until just marked on the other side. Remove from the grill and transfer to baking sheets; repeat with the remaining dough.
Add the toppings: Working on the baking sheets, top the crusts with parmigiano and prosciutto, or top with taleggio and the puttanesca mixture; return to the grill. Crack an egg directly onto each prosciutto pizza, if you want. Cover the grill and cook until the cheese melts and the eggs are set, 2 to 3 minutes.
Finish the pizzas: Remove from the grill and top the prosciutto pizzas with arugula; drizzle with the prepared chile oil. Top the puttanesca pizzas with chives.