Peel the eggplant, leaving 1 inch of skin at the top and bottom. Slice the eggplant lengthwise into 1/4-inch-thick slices (I would use a mandoline for this).
Place the eggplant slices on a cooling rack set over the sink and sprinkle generously with kosher salt. Wait 15 minutes, flip, sprinkle again and wait another 15 minutes. Rinse thoroughly under cool water and gently squeeze out excess water. Place on paper towels and pat dry, then cut the slices into 1/4-inch-wide strips so that they resemble linguine.
Heat a 10-inch saute pan over medium-high heat and add the oil. When it shimmers, add the garlic and red pepper flakes and toss for 10 seconds. Add the eggplant and toss to coat. Add the tomato and toss for 15 to 20 seconds. Add the cream and toss for another 10 seconds. Finish with the basil and Parmesan. Transfer to a serving dish. Top with the breadcrumbs, toss and serve immediately.