Cut tomatoes in half lengthwise. Lay halves cut side up in a single layer in an 8- by 12-inch oval or rectangular casserole (rim should be at least 1/2 in. higher than tomatoes).
In a 1 1/2- to 2-quart pan, combine onion, vinegar, sugar, olive oil, and ginger. Bring to a boil over high heat, stirring often. Pour over tomatoes.
Roast in a 400° oven (convection not recommended) until tomatoes are dark brown and sauce is browned and thickened (bubbles will be large and shiny), about 1 1/4 hours; baste tomatoes with sauce and onions about every 15 minutes at first, then more frequently as mixture begins to thicken, to avoid scorching. Use hot, warm, or at room temperature.
Spread lemon cheese equally on oiled sides of the olive-oil toast slices. Set on plates and cover equally with arugula leaves. Spoon tomatoes and juices equally onto toast slices. Add salt to taste.
Lemon Cheese: In a food processor or bowl, combine 1/3 cup cream cheese (at room temperature), 1/3 cup packed fresh chèvre (goat cheese), 2 tablespoons milk, and 1 teaspoon grated lemon peel. Whirl or beat with a mixer on medium-high speed until blended.
Olive-oil toast. Lay 8 slices of rustic white bread such as sourdough or ciabatta (about 3 by 4 in., cut about 1/2 in. thick) in a single layer in a shallow 10- by 15-inch pan. Bake in a 375° oven until lightly browned on the bottom, 5 to 6 minutes. Turn slices over and brush equally with about 2 tablespoons extra-virgin olive oil. Continue baking until golden brown and crisp, 6 to 8 minutes longer. Use warm, or let cool on a rack.
Nutritional analysis per piece.