2 pounds summer squash (zucchini, yellow squash, pattypan, etc.)
1 tablespoon Kosher salt, plus more for sprinkling
2 tablespoons + 2 teaspoons olive oil, divided
2 shallots, sliced
2 cloves garlic, minced
1/4 teaspoon smoked paprika, plus more for sprinkling
1/2 pound tomatoes, chopped
4 ounces soft goat cheese (chèvre)
1/4 cup loosely packed chiffonade of basil, plus more for garnish
4 eggs
Freshly ground black pepper