Combine the warm water and sugar in a small bowl and stir to dissolve the sugar. Sprinkle the yeast into the water and let stand for about 5 minutes, until creamy.

Combine 3 1/2 cups of the flour, the salt, and the yeast mixture in the bowl of an electric mixer fitted with a paddle attachment. Mix on low speed with a dough hook until the dough comes together and forms a ball, 2 to 3 minutes. Increase the speed to medium and knead with the hook for 5 minutes. If the dough is still sticky, continue mixing and gradually add as much of the remaining 1/2 cup flour as needed.

Oil a large mixing bowl with 2 teaspoons of the oil. Put the dough in the bowl and turn to coat all sides with oil. Cover with plastic wrap or a damp kitchen towel. Let stand in a warm, draft-free place until the dough doubles in size, about 1 1/2 hours.

Grease a 9 x 13-inch baking sheet with 1 tablespoon of the olive oil. Turn the dough out onto the baking sheet and press it evenly over the bottom of the pan. Cover with plastic wrap and set in a warm, draft-free place until it rises to nearly 1 inch thick, about 30 minutes.

Preheat the oven to 400°F.

Make dimples in the dough at 3-inch intervals with your fingertips. Drizzle with 1 tablespoon of the olive oil and bake until just golden brown, 25 minutes.

Remove from the oven and drizzle with 1 tablespoon of the olive oil. Press the grapes into the dough. Return to the oven and bake until lightly golden brown, about 10 minutes. Remove from the oven and sprinkle with the cheese and walnuts. Bake until the cheese melts, about 5 minutes.

Drizzle the focaccia with the remaining 1 tablespoon olive oil. With a long spatula, transfer the bread to a rack and let cool slightly. Cut into squares and serve warm, or let cool to room temperature.