Combine half of the onion, the vinegar and a pinch of salt in a medium microwave-safe bowl. Microwave until hot; set aside.
Heat the olive oil in a large skillet over medium-high heat. Add the remaining onion and cook, stirring, 3 minutes. Add the artichokes and heat through, 4 minutes. Stir in half of the spinach and cook until wilted, 2 more minutes. Remove from the heat and stir in the asiago, goat cheese, mayonnaise and the remaining spinach.
Heat a grill pan over medium heat. Sandwich the spinach-artichoke mixture between the bread slices. Brush the sandwiches with olive oil and add to the grill pan. Grill 3 minutes per side. Let rest 2 minutes; cut in half.
Drain the reserved onion. Drizzle with olive oil, then add the tomatoes, parsley, and salt and pepper to taste; toss. Serve with the sandwiches.