Prepare the farina cakes: Lightly oil a 9×9-inch baking dish and set aside. In a large saucepan, bring water to a boil. Add salt. Whisk in farina, reduce heat to low, and stir briskly to prevent lumps from forming. Cover and cook on low for 10-15 minutes, stirring occasionally. Continue cooking and stirring until the farina is the consistency of a very thick porridge. Pour into the prepared dish, smooth the top, and let stand until no longer steaming, about 10 minutes. Refrigerate, uncovered, until cold and set, about 1-1/2 hours. If not using immediately, cover with plastic wrap and refrigerate until ready to serve (up to 2 days).
When ready to serve: Place a rack in the center of oven and preheat to 200°F. Invert the set farina onto a large cutting board. Cut into squares with a knife or into 3-inch rounds with a biscuit cutter (cutting into squares will yield more cakes—biscuit cutters will leave some scraps.) Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium heat and cook the farina cakes for 3 to 5 minutes, turning once, until browned on the outside and heated inside. Repeat until all of the cakes are seared, adding more butter and oil as needed. Remove to a baking sheet and keep warm in preheated oven until ready to serve.
Prepare the sundried tomato goat cheese sauce: Place sundried tomatoes in a small bowl and cover with boiling water to rehydrate. Let sit 5 minutes, then drain. Warm the milk in the microwave for about 30 seconds. In a blender or a food processor fitted with a steel blade, add the milk, drained sundried tomatoes, goat cheese, mustard, lemon juice, and cayenne. Blend until mostly smooth (the sundried tomatoes will still be a little chunky.)
Prepare the garlic arugula: In a large, deep skillet or Dutch oven, heat the olive oil over medium. Add garlic and sauté for 1 minute. Do not allow the garlic to brown. Add the arugula, salt, and pepper, tossing to coat with garlic and oil. Cover and cook for 2 minutes. Uncover, turn the heat to high, and cook for additional 30 seconds to 1 minute, stirring until the arugula is wilted. Turn off heat, cover to keep warm, and set aside.
Poach the eggs: Fill a deep saucepot with 1 inch of water and add vinegar. Heat water until just below the simmering point (small bubbles will appear all over the bottom of the pan but will not break the water’s surface.) Reduce heat slightly to keep the water from simmering. Break the first egg into a small dish. Using a rubber spatula, move in a vigorous circular motion around the sides of the pot to create a “whirlpool.” Gently slide the egg from the dish into the whirlpool’s center. Do not worry if the egg appears to be coming apart—the motion of the water will draw the sides in and around. If your egg sticks to the bottom of the pot, wait 30 seconds then gently slide the spatula underneath to loosen it. Cook egg for 3-4 minutes, until it only jiggles a little when nudged with the spatula. With a slotted spoon, carefully remove to a paper-towel lined plate. Repeat with remaining eggs. Just before serving, you can slip the eggs back into the warm (not simmering) water briefly to rewarm them if desired.
Serve: Top each farina cake with a spoonful of sauce, the sautéed arugula, a poached egg, and then another spoonful of sauce. Serve immediately.