Finely grate the lemon to yield 1 tsp. zest and juice it to yield 1-1/2 tsp. juice. In a small bowl, combine the zest, juice, vinegar, maple syrup, and 1 Tbs. water.
In a 5- to 6-quart Dutch oven (or other deep, wide pan), heat 1 Tbs. of the butter and the olive oil over low heat. Add the carrots and 3/4 tsp. salt. Cover and cook, stirring frequently but gently, until the carrots are nicely browned and just tender, about 20 minutes. With a slotted spoon, transfer the carrots to a large plate.
Add 1 Tbs. butter to the remaining fat in the pan. (If there’s no fat in the pan, add 1 Tbs. olive oil too.) When the butter has melted, arrange the fingerlings cut side down in a single layer in the pan and season with 3/4 tsp. salt. Cover partially and cook, undisturbed, until the potatoes are deep golden-brown on the bottom, 5 to 7 minutes. Add the chicken broth or water and bring to a boil; reduce to a simmer and cover partially. Cook until the potatoes are tender and the liquid has reduced to 2 to 3 Tbs., 12 to 14 minutes.
Add the garlic to the potatoes and cook, stirring very gently, until fragrant, about 30 seconds. Add the reserved carrots and the peas, spinach, and lemon juice mixture. Stir gently until the spinach is wilted, 1 to 2 minutes. Remove the pan from the heat and stir in the remaining 1-1/2 Tbs. butter until just melted. Stir in the tarragon. Transfer the vegetables to a platter and serve.