Sauté the Aromatics: Heat 1 tablespoon oil in a large skillet over medium heat. Add the shallots, garlic, and 2 teaspoons thyme. Cook for about 3 minutes, stirring occasionally, until the shallots begin to soften.
Add the Wine: Pour the dry white wine into the skillet. Stir and scrape the bottom of the skillet to loosen any browned pieces.
Cook the Tomatoes: Add the chopped tomatoes, ¼ teaspoon salt, and ¼ teaspoon black pepper. Cook over medium heat for about 5 minutes, stirring occasionally, until the tomatoes soften and form a chunky sauce. Reduce the heat to low to keep the topping warm.
Preheat the Waffle Iron: Preheat the waffle iron to medium-high heat while preparing the waffle batter.
Combine the Dry Ingredients: Place the chickpea flour, baking powder, remaining ¼ teaspoon salt, and remaining ¼ teaspoon black pepper in a large mixing bowl. Whisk until evenly combined.
Add the Wet Ingredients: Add the milk, 1 tablespoon oil, egg yolk, and vinegar to the dry ingredients. Whisk until the batter is smooth and no dry pockets remain.
Add the Cheese: Stir the shredded white cheddar into the batter until evenly distributed.
Whip the Egg White: Place the egg white in a clean medium mixing bowl. Beat it with an electric hand mixer for 2 to 3 minutes, or until stiff peaks form.
Fold in the Egg White: Add the whipped egg white to the chickpea batter. Gently fold it in with a rubber spatula until just combined. Avoid overmixing so the batter stays light.
Cook the Waffles: Spoon the batter into the preheated waffle iron without overfilling it. Close the waffle iron and cook for about 5 minutes, or until the waffles are crisp, golden brown, and release easily. Cook in batches if necessary.
Add the Tomato Topping: Divide the cooked waffles between two plates. Spoon the warm tomato and shallot mixture evenly over the waffles.
Finish the Plates: Place one poached egg on top of each serving. Sprinkle with the remaining fresh thyme and serve immediately.