Herb-Infused Lemon-Strawberry Loaf

Preheat oven to 350°F. Butter an 8 1/2x4 1/2" loaf pan, then line with parchment paper, leaving a 2" overhang on 2 sides.
Lightly crush herbs with your hands to help release oils, then transfer to a small saucepan. Add 3/4 cup butter and cook over medium heat, swirling pan often, until butter melts, about 5 minutes. Let cool to room temperature, 15–20 minutes.
Meanwhile, stir 3/4 cup milk, 1 Tbsp. lemon zest, and 1/4 cup lemon juice in a small bowl. Whisk flour, baking powder, and salt in a medium bowl.
Using an electric mixer on medium speed (or a stand mixer fitted with the whisk attachment), beat eggs and granulated sugar in a large bowl until light in color and doubled in size, 4–5 minutes. Discard herbs from butter, then beat butter into egg mixture until well combined. Add half of dry ingredients and beat just until combined. Beat in milk mixture, then add remaining dry ingredients and beat just until combined. Do not overmix.
Pour half of batter into prepared pan; smooth top. Top with 3 Tbsp. jam and swirl through batter with a butter knife or chopstick. Pour remaining batter over; smooth top. Repeat swirling process with remaining 3 Tbsp. jam.
Bake cake until a tester inserted into the center comes out with moist crumbs attached, 50–60 minutes.
Transfer pan to a wire rack and let cool 20 minutes. Remove cake from pan using parchment overhang and let cool completely.
Whisk powdered sugar and remaining 1 Tbsp. milk and 1 tsp. lemon zest in a small bowl until smooth. Spoon over top of cake and let drip down sides.
Do Ahead: Cake can be baked 3 days ahead. Store in an airtight container at room temperature, or freeze up to 3 months.