Stir first 6 ingredients in bowl until smooth. Season filling with pepper. Place 1 phyllo sheet on work surface (keep remainder covered). Brush lightly with oil and season with pepper. Top with 1 more phyllo sheet. Brush lightly with oil and season with pepper. Repeat with 1 more sheet. Fold stacked phyllo in half lengthwise. Brush top with oil.
Spoon 1/4 of filling in 1-inch-wide log down 1 long side of sheet, leaving 1-inch borders. Fold each short end over filling. Brush edges with oil. Press seam to seal. Wrap strudel tightly in plastic. Refrigerate seam side down. Repeat with remaining phyllo and filling, forming 4 strudels total. (Can be prepared 2 days ahead.)
Preheat oven to 375 degrees. Lightly oil cookie sheets. Place strudels on prepared sheets, seam sides down. Using serrated knife, score each strudel (cutting through phyllo only), making 14 diagonal cuts in each. Brush strudels with oil. Bake until golden, about 15 minutes. Cool 10 minutes.
Cut through score lines, forming slices. Arrange cut side up on platter. Garnish slices with tomato and oregano.