MAKE THE DOUGH: In the bowl of an electric stand mixer fitted with the dough hook, combine the yeast with the water and sugar and let stand until slightly foamy, about 5 minutes. Add the egg, salt, olive oil, and the 1 3/4 cups of flour and beat at medium speed until a smooth dough forms, about 5 minutes. Knead the dough by hand for 1 minute, then transfer it to an oiled bowl. Cover the bowl with plastic wrap and let rest until the dough has doubled in bulk, about 30 minutes.

Preheat the oven to 400 degrees. Punch down the flatbread dough and divide it into 4 pieces. Pinch each piece of dough into a ball. Dust the balls with flour, cover with plastic wrap and let rest for 20 minutes.

MEANWHILE, MAKE THE TOPPING: Place the tomatoes on a rimmed baking sheet, cut side down. Scatter the garlic and thyme all around them and drizzle with 2 tablespoons of the olive oil. Roast in the center of the oven for about 15 minutes, until slightly dry.

On a lightly floured surface, roll out each dough ball to an 8-inch round and brush the flatbread with the remaining 4 teaspoons of olive oil. Heat a large griddle until very hot. Add 2 of the dough rounds and cook over high heat until the bottoms are browned and set and the top bubbles slightly, about 2 minutes. Transfer the flatbreads to a large baking sheet, cooked side down. Repeat with the remaining dough rounds.

Spread the ricotta on the flatbreads and top with the tomatoes and anchovies. Season with pepper. Bake for about 15 minutes, until the flatbreads are lightly browned around the edges. Transfer the flatbreads to a work surface and lightly drizzle with balsamic vinegar. Scatter the basil on top. Slice the flatbreads into wedges and serve right away.

The flatbread dough can be refrigerated overnight. Bring the dough to room temperature before proceeding. The flatbreads can be baked early in the day and reheated before serving.