Brush 1 side of each bread slice with 1/2 teaspoon oil. Spread 1/2 ounce goat cheese on other side of each slice. Layer beet, carrots, pear, and arugula on cheese side of 4 slices; drizzle with lemon juice, season with salt and pepper, and sandwich with remaining slices, cheese side down.
Heat a large nonstick skillet over medium. In two batches, cook sandwiches, pressing gently on tops with a spatula, until golden brown, about 3 minutes per side.