Summer Squash With Eggs

Ingredients

• 2 lbs summer squash, unpeeled (zucchini, yellow or crookneck squash, pattypan, etc.)
• 1 tablespoon kosher salt, plus more for sprinkling (I used French grey salt instead)
• 2 tablespoons olive oil, divided
• 2 teaspoons olive oil, divided
• 2 shallots, sliced
• 2 garlic cloves, minced
• 1/4 teaspoon smoked paprika, plus more for sprinkling
• 1/2 lb tomatoes, chopped (I used a variety of heirloom tomatoes from the farmer's market-the very small ones I left whole)
• 4 ounces soft fresh goat cheese (chevre, for example)
• 1/4 cup loosely packed chiffonade basil, plus more for garnish
• 4 eggs
• fresh cracked black pepper